נדרים עט.
"קיים בלבו קיים. הפר בלבו אינו מופר"
The [יו"ד רל"ד, מא] שו"ע paskens that a husband may be מקיים his wife’s neder בלבו. But if the בעל is מפר the neder בלבו the הפרה is not חל. The ר"ן explains that the בעל does not need to be "מוציא בשפותיו" the הקמה. This is derived from the halacha of "שתיקה ביום שמעו". That if the husband heard the neder and was quiet (שתק) the entire day, when the day is over the נדר is considered מקויים. This is because since he didn’t use his opportunity to be מיפר it is apparent that he approved the neder. So, we see from here that approval (הקמה) even if not spoken out, is a good הקמה. So too, a הקמה בלב works immediately and is considered a good הקמה.
The source for the halacha that normally "דברים שבלב לא הוי דברים" is in קידושין (מט:). There the Gemara brings a story of someone in חוץ לארץ planning to move to ארץ ישראל, and he sold off all of his possessions. After the sale, his plans fell through so he wanted to void the sale since the only reason for the sale was because he was moving to א"י. Rava paskened that even though we knew about his plans, since at the time of the sale he didn’t state clearly his intentions (אדעתא דהכי), they therefore just remain דברים שבלב and the sale stands. The רשב"א explains that this remains דברים שבלב even though we believe what his intentions were, since at the time of the sale his intentions were not "מוכח" (apparent) and people sometimes do change their minds, thus, his intentions remain דברים שבלב and they cannot affect the sale.
Also, [גליון הש"ס כאן] רע"א asks on the above ר"ן. Even though we normally hold "דברים שבלב לא הוי דברים", if your thoughts are obvious (מוכח), then although not expressed explicitly they are considered דברים and taken into account. If so, asks רע"א, the reason why "שתיקה ביום שמעו" is a good הקמה, is because by the end of the day it becomes apparent (אנן סהדי) and מוכח that the בעל was מיקם with his שתיקה so there is no חסרון of דברים שבלב. If so, how can we learn out from this that if the בעל makes a הקמה בלב - it should be an immediate הקמה? Since all this is happening in his heart, there is nothing מוכח or apparent telling us that he wants to be מקים! So how can this be learned from the דיןof שתיקה? He bleibs with a צע"ג.
The אחרונים discuss why we have to learn "שתיקה ביום שמעו" is as if he made an actual הקמה. Why can’t we say that since the Torah gives a time limit to make a הפרה, which is the day he hears about it and afterwards he loses the opportunity and the neder stands, how do we know it’s because of a גילוי דעת that he wants the neder? It’s valid because now the time period for allowing a הפרה has elapsed!
The פירוש הרא"ש says that when one is מקים בלב he says "קיים ליכי" in his heart. Some learn that the רא"ש means to say that one must literally think these words in his heart.