Theorem bicom 139

Description: Commutative law for equivalence. Theorem *4.21 of [WhiteheadRussell] p. 117. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 11-Nov-2012.)

Assertion

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

Proof of Theorem bicom

Step	Hyp	Ref	Expression
1		bicom1 130	. 2 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 ↔ 𝜑))
2		bicom1 130	. 2 ⊢ ((𝜓 ↔ 𝜑) → (𝜑 ↔ 𝜓))
3	1, 2	impbii 125	1 ⊢ ((𝜑 ↔ 𝜓) ↔ (𝜓 ↔ 𝜑))

Syntax hints:   ↔ wb 104

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

This theorem is referenced by:  bicomd  140  bibi1i  227  bibi1d  232  ibibr  245  bibif  688  con2bidc  861  con2biddc  866  pm5.17dc  890  bigolden  940  nbbndc  1373  bilukdc  1375  falbitru  1396  3impexpbicom  1415  exists1  2096  eqcom  2142  abeq1  2250  necon2abiddc  2375  necon2bbiddc  2376  necon4bbiddc  2383  ssequn1  3251  axpow3  4109  isocnv  5720  suplocsrlem  7640  uzennn  10240  bezoutlemle  11732