Theorem opabbii 3853

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 2082	. 2 |- z = z
2		opabbii.1	. . . 4 |- ( ph <-> ps )
3	2	a1i 9	. . 3 |- ( z = z -> ( ph <-> ps ) )
4	3	opabbidv 3852	. 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 1285   {copab 3846

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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-opab 3848

This theorem is referenced by:  mptv  3882  fconstmpt  4413  xpundi  4422  xpundir  4423  inxp  4498  cnvco  4548  resopab  4682  opabresid  4689  cnvi  4758  cnvun  4759  cnvin  4761  cnvxp  4772  cnvcnv3  4800  coundi  4852  coundir  4853  mptun  5060  fvopab6  5296  cbvoprab1  5607  cbvoprab12  5609  dmoprabss  5617  mpt2mptx  5626  resoprab  5628  ov6g  5669  dfoprab3s  5847  dfoprab3  5848  dfoprab4  5849  xpcomco  6370  dmaddpq  6631  dmmulpq  6632  recmulnqg  6643  enq0enq  6683  ltrelxr  7240  ltxr  8927  shftidt2  9858