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	|- ( ph <-> ps )

Assertion

Ref	Expression
bicomi	|- ( ps <-> ph )

Proof of Theorem bicomi

Step	Hyp	Ref	Expression
1		bicomi.1	. 2 |- ( ph <-> ps )
2		bicom1 131	. 2 |- ( ( ph <-> ps ) -> ( ps <-> ph ) )
3	1, 2	ax-mp 5	1 |- ( ps <-> ph )

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  559  anabs1  574  anabs7  576  an43  590  pm4.76  608  mtbir  678  sylnibr  684  sylnbir  686  xchnxbir  688  xchbinxr  690  nbn  707  pm4.25  766  pm4.56  788  pm4.77  807  pm3.2an3  1203  syl3anbr  1318  3an6  1359  truan  1415  truimfal  1455  nottru  1458  sbid  1823  sb10f  2049  cleljust  2209  eqabdv  2363  nfabdw  2403  necon3bbii  2449  rspc2gv  2932  alexeq  2942  ceqsrexbv  2947  clel2  2949  clel4  2952  dfsbcq2  3044  cbvreucsf  3202  dfdif3  3328  raldifb  3358  difab  3489  un0  3541  in0  3542  ss0b  3547  rexdifpr  3716  snssb  3826  snssg  3827  iindif2m  4058  epse  4462  abnex  4567  uniuni  4571  elco  4920  cotr  5143  issref  5144  mptpreima  5255  ralrnmpt  5818  rexrnmpt  5819  eroveu  6859  mapsnend  7051  wrd2ind  11408  fprodseq  12262  issrg  14098  toptopon  14870  xmeterval  15287  txmetcnp  15370  dedekindicclemicc  15484  eldvap  15534  fsumdvdsmul  15846  isclwwlk  16376  iseupthf1o  16430  eupth2lem1  16440  bdeq  16580  bd0r  16582  bdcriota  16640  bj-d0clsepcl  16682  bj-dfom  16690