Theorem truan 1548

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 1541	. . 3 ⊢ ⊤
2	1	biantrur 530	. 2 ⊢ (𝜑 ↔ (⊤ ∧ 𝜑))
3	2	bicomi 224	1 ⊢ ((⊤ ∧ 𝜑) ↔ 𝜑)

Syntax hints:   ↔ wb 206   ∧ wa 395  ⊤wtru 1538

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 207  df-an 396  df-tru 1540

This theorem is referenced by:  truanfal  1571  euelss  4351  tgcgr4  28557  aciunf1  32681  sgn3da  34506  wl-2mintru1  37456  truconj  38061  tradd  38065  ifpdfxor  43449  dfid7  43574  eel0TT  44675  eelT00  44676  eelTTT  44677  eelT11  44678  eelT12  44680  eelTT1  44681  eelT01  44682  eel0T1  44683  eelTT  44742  uunT1p1  44752  uunTT1  44764  uunTT1p1  44765  uunTT1p2  44766  uunT11  44767  uunT11p1  44768  uunT11p2  44769  uunT12  44770  uunT12p1  44771  uunT12p2  44772  uunT12p3  44773  uunT12p4  44774  uunT12p5  44775