Theorem rabbiia 2547

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 427	. . 3 |- ( ( x e. A /\ ph ) <-> ( x e. A /\ ps ) )
3	2	abbii 2153	. 2 |- { x | ( x e. A /\ ph ) } = { x | ( x e. A /\ ps ) }
4		df-rab 2315	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
5		df-rab 2315	. 2 |- { x e. A | ps } = { x | ( x e. A /\ ps ) }
6	3, 4, 5	3eqtr4i 2070	1 |- { x e. A | ph } = { x e. A | ps }

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   e. wcel 1393   {cab 2026   {crab 2310

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-rab 2315

This theorem is referenced by:  bm2.5ii  4222  fndmdifcom  5273  cauappcvgprlemladdru  6754  cauappcvgprlemladdrl  6755  cauappcvgpr  6760  caucvgprlemcl  6774  caucvgprlemladdrl  6776  caucvgpr  6780  caucvgprprlemclphr  6803  ioopos  8819