Theorem opabbii 3845

Description: Equivalent wff's yield equal class abstractions. (Contributed by NM, 15-May-1995.)

Hypothesis

Ref	Expression
opabbii.1	|- ( ph <-> ps )

Assertion

Ref	Expression
opabbii	|- { <. x , y >. | ph } = { <. x , y >. | ps }

Proof of Theorem opabbii

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2081	. 2 |- z = z
2		opabbii.1	. . . 4 |- ( ph <-> ps )
3	2	a1i 9	. . 3 |- ( z = z -> ( ph <-> ps ) )
4	3	opabbidv 3844	. 2 |- ( z = z -> { <. x , y >. | ph } = { <. x , y >. | ps } )
5	1, 4	ax-mp 7	1 |- { <. x , y >. | ph } = { <. x , y >. | ps }

Syntax hints:   <-> wb 103   = wceq 1284   {copab 3838

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-opab 3840

This theorem is referenced by:  mptv  3874  fconstmpt  4405  xpundi  4414  xpundir  4415  inxp  4488  cnvco  4538  resopab  4672  opabresid  4679  cnvi  4748  cnvun  4749  cnvin  4751  cnvxp  4762  cnvcnv3  4790  coundi  4842  coundir  4843  mptun  5049  fvopab6  5285  cbvoprab1  5596  cbvoprab12  5598  dmoprabss  5606  mpt2mptx  5615  resoprab  5617  ov6g  5658  dfoprab3s  5836  dfoprab3  5837  dfoprab4  5838  xpcomco  6323  dmaddpq  6569  dmmulpq  6570  recmulnqg  6581  enq0enq  6621  ltrelxr  7173  ltxr  8849  shftidt2  9720