Theorem trut 1545

Description: A proposition is equivalent to it being implied by ⊤. Closed form of mptru 1546. Dual of dfnot 1558. It is to tbtru 1547 what a1bi 363 is to tbt 370. (Contributed by BJ, 26-Oct-2019.)

Assertion

Ref	Expression
trut	⊢ (𝜑 ↔ (⊤ → 𝜑))

Proof of Theorem trut

Step	Hyp	Ref	Expression
1		tru 1543	. 2 ⊢ ⊤
2	1	a1bi 363	1 ⊢ (𝜑 ↔ (⊤ → 𝜑))

Syntax hints:   → wi 4   ↔ wb 205  ⊤wtru 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-tru 1542

This theorem is referenced by:  truimfal  1563  euae  2661