Theorem bicom 222

Description: Commutative law for the biconditional. Theorem *4.21 of [WhiteheadRussell] p. 117. (Contributed by NM, 11-May-1993.)

Assertion

Ref	Expression
bicom	⊢ ((𝜑 ↔ 𝜓) ↔ (𝜓 ↔ 𝜑))

Proof of Theorem bicom

Step	Hyp	Ref	Expression
1		bicom1 221	. 2 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 ↔ 𝜑))
2		bicom1 221	. 2 ⊢ ((𝜓 ↔ 𝜑) → (𝜑 ↔ 𝜓))
3	1, 2	impbii 209	1 ⊢ ((𝜑 ↔ 𝜓) ↔ (𝜓 ↔ 𝜑))

Syntax hints:   ↔ wb 206

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

This theorem is referenced by:  bicomd  223  bibi1i  338  bibi1d  343  con2bi  353  ibibr  368  bibif  371  nbbn  383  biluk  385  biadan  819  pm5.17  1014  bigolden  1029  xorcom  1516  norass  1539  trubifal  1573  hadcoma  1601  eqabcbw  2811  eqabcb  2877  pm13.183  3622  ssequn1  4140  isocnv  7288  qextlt  13132  qextle  13133  rpnnen2lem12  16164  odd2np1  16282  sumodd  16329  nrmmetd  24535  lgsqrmodndvds  27337  eqelbid  32567  mgccnv  33098  cvmlift2lem12  35536  nn0prpw  36545  wl-3xorrot  37759  wl-3xorcoma  37760  tsbi4  38416  bicomdd  39259  onsupmaxb  43625  ifpbicor  43860  rp-fakeoranass  43899  or3or  44408  3impexpbicom  44865  3impexpbicomVD  45241  notbicom  45553  limsupreuz  46124  nabctnabc  47320