Many have the practice both in speech and in writing to mention Gd or heavenly help with the use of the expressions be’ezrat hashem, im yirtzeh hashem or biseyata dishmaya.
What is the source for this?
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Citing from Wikipedia:
The book "Toldot Yitzhak" (‘The Offspring of Isaac’), by Yitzhak Karo, Yosef Karo's uncle, offers the meaning of this custom of writing ב"ה (B"H), at the top of every letter, with accordance to the biblical verse: "In all thy ways acknowledge Him, and He will direct thy paths" (Book of Proverbs 3:6)."
While the source mentions Baruch Hashem, it seems logical that the other expressions are said or written for the same reason, emanating from the same Proverbs verse.
Eliezer and Yitro also used the phrase Baruch Hashem.
With regards to saying "אם ירצה ה" - 'G-d willing' the Shlah HaKodesh writes in Shaar Osiyos, Emes V'Emunah the following:
בכלל דביקות האמונה שיאמר על כל פעולה שרוצה לעשות אפילו לזמן קרוב יאמר אעשה זה אם ירצה השם, הן שיהיה דבר גדול או דבר קטן. וסימנך (משלי יט, כא) עצת ה' היא תקום. היא ראשי תיבות אם ירצה השם, ואז תקום
Regarding the principle of the clinging of faith that one should say on every action that he wants to do, even for something in the near future, he should say 'G-d willing', whether it be for a big thing or a small thing. And the siman for this is, (Proverbs 19:211) "But it is the LORD’s plan" (וַעֲצַ֥ת ה' הִ֣יא) - the word "הִ֣יא" is roshei teivos (i.e. the letters spell out) אם ירצה השם - (when this is on the mind) then "that is accomplished".
1 The pasuk writes: רַבּ֣וֹת מַחֲשָׁב֣וֹת בְּלֶב־אִ֑ישׁ וַעֲצַ֥ת ה' הִ֣יא תָקֽוּם - Many designs are in a man’s mind, But it is the LORD’s plan that is accomplished.