Theorem truan 1260

Description: True can be removed from a conjunction. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Wolf Lammen, 21-Jul-2019.)

Assertion

Ref	Expression
truan	⊢ ((⊤ ∧ 𝜑) ↔ 𝜑)

Proof of Theorem truan

Step	Hyp	Ref	Expression
1		tru 1247	. . 3 ⊢ ⊤
2	1	biantrur 287	. 2 ⊢ (𝜑 ↔ (⊤ ∧ 𝜑))
3	2	bicomi 123	1 ⊢ ((⊤ ∧ 𝜑) ↔ 𝜑)

Syntax hints:   ∧ wa 97   ↔ wb 98  ⊤wtru 1244

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110  df-tru 1246

This theorem is referenced by:  truanfal  1293  truxortru  1310  truxorfal  1311