Theorem opabbii 4152

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 2229	. 2 |- z = z
2		opabbii.1	. . . 4 |- ( ph <-> ps )
3	2	a1i 9	. . 3 |- ( z = z -> ( ph <-> ps ) )
4	3	opabbidv 4151	. 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 1395   {copab 4145

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-opab 4147

This theorem is referenced by:  mptv  4182  fconstmpt  4769  xpundi  4778  xpundir  4779  inxp  4860  cnvco  4911  resopab  5053  opabresid  5062  cnvi  5137  cnvun  5138  cnvin  5140  cnvxp  5151  cnvcnv3  5182  coundi  5234  coundir  5235  mptun  5459  fvopab6  5737  cbvoprab1  6086  cbvoprab12  6088  dmoprabss  6096  mpomptx  6105  resoprab  6110  ov6g  6153  dfoprab3s  6346  dfoprab3  6347  dfoprab4  6348  opabn1stprc  6351  mapsncnv  6857  xpcomco  7003  dmaddpq  7587  dmmulpq  7588  recmulnqg  7599  enq0enq  7639  ltrelxr  8228  ltxr  9998  shftidt2  11380  prdsex  13339  prdsval  13343  prdsbaslemss  13344  releqgg  13794  eqgex  13795  dvdsrzring  14604  lmfval  14904  lmbr  14924  cnmptid  14992  lgsquadlem3  15795  wksfval  16110