Theorem opabbii 4127

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 2207	. 2 |- z = z
2		opabbii.1	. . . 4 |- ( ph <-> ps )
3	2	a1i 9	. . 3 |- ( z = z -> ( ph <-> ps ) )
4	3	opabbidv 4126	. 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 1373   {copab 4120

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-opab 4122

This theorem is referenced by:  mptv  4157  fconstmpt  4740  xpundi  4749  xpundir  4750  inxp  4830  cnvco  4881  resopab  5022  opabresid  5031  cnvi  5106  cnvun  5107  cnvin  5109  cnvxp  5120  cnvcnv3  5151  coundi  5203  coundir  5204  mptun  5427  fvopab6  5699  cbvoprab1  6040  cbvoprab12  6042  dmoprabss  6050  mpomptx  6059  resoprab  6064  ov6g  6107  dfoprab3s  6299  dfoprab3  6300  dfoprab4  6301  mapsncnv  6805  xpcomco  6946  dmaddpq  7527  dmmulpq  7528  recmulnqg  7539  enq0enq  7579  ltrelxr  8168  ltxr  9932  shftidt2  11258  prdsex  13216  prdsval  13220  prdsbaslemss  13221  releqgg  13671  eqgex  13672  dvdsrzring  14480  lmfval  14779  lmbr  14800  cnmptid  14868  lgsquadlem3  15671