Theorem rabbiia 2715

Description: Equivalent wff's yield equal restricted class abstractions (inference form). (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 451	. . 3 |- ( ( x e. A /\ ph ) <-> ( x e. A /\ ps ) )
3	2	abbii 2286	. 2 |- { x | ( x e. A /\ ph ) } = { x | ( x e. A /\ ps ) }
4		df-rab 2457	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
5		df-rab 2457	. 2 |- { x e. A | ps } = { x | ( x e. A /\ ps ) }
6	3, 4, 5	3eqtr4i 2201	1 |- { x e. A | ph } = { x e. A | ps }

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   {cab 2156   {crab 2452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-rab 2457

This theorem is referenced by:  rabbii  2716  bm2.5ii  4480  fndmdifcom  5602  cauappcvgprlemladdru  7618  cauappcvgprlemladdrl  7619  cauappcvgpr  7624  caucvgprlemcl  7638  caucvgprlemladdrl  7640  caucvgpr  7644  caucvgprprlemclphr  7667  ioopos  9907  gcdcom  11928  gcdass  11970  lcmcom  12018  lcmass  12039