Theorem rabbiia 2671

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 449	. . 3 |- ( ( x e. A /\ ph ) <-> ( x e. A /\ ps ) )
3	2	abbii 2255	. 2 |- { x | ( x e. A /\ ph ) } = { x | ( x e. A /\ ps ) }
4		df-rab 2425	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
5		df-rab 2425	. 2 |- { x e. A | ps } = { x | ( x e. A /\ ps ) }
6	3, 4, 5	3eqtr4i 2170	1 |- { x e. A | ph } = { x e. A | ps }

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   {cab 2125   {crab 2420

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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-rab 2425

This theorem is referenced by:  rabbii  2672  bm2.5ii  4412  fndmdifcom  5526  cauappcvgprlemladdru  7464  cauappcvgprlemladdrl  7465  cauappcvgpr  7470  caucvgprlemcl  7484  caucvgprlemladdrl  7486  caucvgpr  7490  caucvgprprlemclphr  7513  ioopos  9733  gcdcom  11662  gcdass  11703  lcmcom  11745  lcmass  11766