Theorem truan 1365

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 1352	. . 3 ⊢ ⊤
2	1	biantrur 301	. 2 ⊢ (𝜑 ↔ (⊤ ∧ 𝜑))
3	2	bicomi 131	1 ⊢ ((⊤ ∧ 𝜑) ↔ 𝜑)

Syntax hints:   ∧ wa 103   ↔ wb 104  ⊤wtru 1349

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116  df-tru 1351

This theorem is referenced by:  truanfal  1397  truxortru  1414  truxorfal  1415