Theorem bj-trut 31740

Description: A proposition is equivalent to it being implied by ⊤. Closed form of trud 1484 (which it can shorten); dual of dfnot 1493. It is to tbtru 1485 what a1bi 351 is to tbt 358, and this appears in their respective proofs. (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-trut	⊢ (𝜑 ↔ (⊤ → 𝜑))

Proof of Theorem bj-trut

Step	Hyp	Ref	Expression
1		tru 1479	. 2 ⊢ ⊤
2	1	a1bi 351	1 ⊢ (𝜑 ↔ (⊤ → 𝜑))

Syntax hints:   → wi 4   ↔ wb 195  ⊤wtru 1476

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 196  df-tru 1478

This theorem is referenced by: (None)