MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  an21 Structured version   Visualization version   GIF version

Theorem an21 643
Description: Swap two conjuncts. (Contributed by Peter Mazsa, 18-Sep-2022.)
Assertion
Ref Expression
an21 (((𝜑𝜓) ∧ 𝜒) ↔ (𝜓 ∧ (𝜑𝜒)))

Proof of Theorem an21
StepHypRef Expression
1 biid 261 . . 3 ((𝜑𝜒) ↔ (𝜑𝜒))
21bianassc 642 . 2 ((𝜓 ∧ (𝜑𝜒)) ↔ ((𝜑𝜓) ∧ 𝜒))
32bicomi 223 1 (((𝜑𝜓) ∧ 𝜒) ↔ (𝜓 ∧ (𝜑𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395
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-an 396
This theorem is referenced by:  an32  645  an13  646  indifdi  4279  fncnv  6620  mpocurryd  8268  rexuz2  12907  resmndismnd  18753  imasabl  19824  logfac2  27143  ltgov  28394  brimg  35527  eldmqsres  37753  xrninxp2  37859
  Copyright terms: Public domain W3C validator