Theorem rabbiia 2591

Description: Equivalent wff's yield equal restricted class abstractions (inference rule). (Contributed by NM, 22-May-1999.)

Hypothesis

Ref	Expression
rabbiia.1	|- ( x e. A -> ( ph <-> ps ) )

Assertion

Ref	Expression
rabbiia	|- { x e. A | ph } = { x e. A | ps }

Proof of Theorem rabbiia

Step	Hyp	Ref	Expression
1		rabbiia.1	. . . 4 |- ( x e. A -> ( ph <-> ps ) )
2	1	pm5.32i 441	. . 3 |- ( ( x e. A /\ ph ) <-> ( x e. A /\ ps ) )
3	2	abbii 2194	. 2 |- { x | ( x e. A /\ ph ) } = { x | ( x e. A /\ ps ) }
4		df-rab 2357	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
5		df-rab 2357	. 2 |- { x e. A | ps } = { x | ( x e. A /\ ps ) }
6	3, 4, 5	3eqtr4i 2111	1 |- { x e. A | ph } = { x e. A | ps }

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   {cab 2067   {crab 2352

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-rab 2357

This theorem is referenced by:  bm2.5ii  4240  fndmdifcom  5294  cauappcvgprlemladdru  6846  cauappcvgprlemladdrl  6847  cauappcvgpr  6852  caucvgprlemcl  6866  caucvgprlemladdrl  6868  caucvgpr  6872  caucvgprprlemclphr  6895  ioopos  8973  gcdcom  10365  gcdass  10404  lcmcom  10446  lcmass  10467