Theorem opabbii 4100

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 2196	. 2 |- z = z
2		opabbii.1	. . . 4 |- ( ph <-> ps )
3	2	a1i 9	. . 3 |- ( z = z -> ( ph <-> ps ) )
4	3	opabbidv 4099	. 2 |- ( z = z -> { <. x , y >. | ph } = { <. x , y >. | ps } )
5	1, 4	ax-mp 5	1 |- { <. x , y >. | ph } = { <. x , y >. | ps }

Syntax hints:   <-> wb 105   = wceq 1364   {copab 4093

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-opab 4095

This theorem is referenced by:  mptv  4130  fconstmpt  4710  xpundi  4719  xpundir  4720  inxp  4800  cnvco  4851  resopab  4990  opabresid  4999  cnvi  5074  cnvun  5075  cnvin  5077  cnvxp  5088  cnvcnv3  5119  coundi  5171  coundir  5172  mptun  5389  fvopab6  5658  cbvoprab1  5994  cbvoprab12  5996  dmoprabss  6004  mpomptx  6013  resoprab  6018  ov6g  6061  dfoprab3s  6248  dfoprab3  6249  dfoprab4  6250  mapsncnv  6754  xpcomco  6885  dmaddpq  7446  dmmulpq  7447  recmulnqg  7458  enq0enq  7498  ltrelxr  8087  ltxr  9850  shftidt2  10997  prdsex  12940  releqgg  13350  eqgex  13351  dvdsrzring  14159  lmfval  14428  lmbr  14449  cnmptid  14517  lgsquadlem3  15320