Theorem bicomi 132

Description: Inference from commutative law for logical equivalence. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 16-Sep-2013.)

Hypothesis

Ref	Expression
bicomi.1	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
bicomi	⊢ (𝜓 ↔ 𝜑)

Proof of Theorem bicomi

Step	Hyp	Ref	Expression
1		bicomi.1	. 2 ⊢ (𝜑 ↔ 𝜓)
2		bicom1 131	. 2 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 ↔ 𝜑))
3	1, 2	ax-mp 5	1 ⊢ (𝜓 ↔ 𝜑)

Syntax hints:   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  biimpri  133  bitr2i  185  bitr3i  186  bitr4i  187  bitr3id  194  bitr3di  195  bitr4di  198  bitr4id  199  pm5.41  251  anidm  396  an21  471  pm4.87  557  anabs1  572  anabs7  574  pm4.76  604  mtbir  671  sylnibr  677  sylnbir  679  xchnxbir  681  xchbinxr  683  nbn  699  pm4.25  758  pm4.56  780  pm4.77  799  pm3.2an3  1176  syl3anbr  1282  3an6  1322  truan  1370  truimfal  1410  nottru  1413  sbid  1774  sb10f  1995  cleljust  2154  nfabdw  2338  necon3bbii  2384  rspc2gv  2855  alexeq  2865  ceqsrexbv  2870  clel2  2872  clel4  2875  dfsbcq2  2967  cbvreucsf  3123  dfdif3  3247  raldifb  3277  difab  3406  un0  3458  in0  3459  ss0b  3464  rexdifpr  3622  snssb  3727  snssg  3728  iindif2m  3956  epse  4344  abnex  4449  uniuni  4453  elco  4795  cotr  5012  issref  5013  mptpreima  5124  ralrnmpt  5660  rexrnmpt  5661  eroveu  6628  fprodseq  11593  issrg  13153  toptopon  13603  xmeterval  14020  txmetcnp  14103  dedekindicclemicc  14195  eldvap  14236  if0ab  14642  bdeq  14660  bd0r  14662  bdcriota  14720  bj-d0clsepcl  14762  bj-dfom  14770