Theorem rabbiia 2722

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 454	. . 3 |- ( ( x e. A /\ ph ) <-> ( x e. A /\ ps ) )
3	2	abbii 2293	. 2 |- { x | ( x e. A /\ ph ) } = { x | ( x e. A /\ ps ) }
4		df-rab 2464	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
5		df-rab 2464	. 2 |- { x e. A | ps } = { x | ( x e. A /\ ps ) }
6	3, 4, 5	3eqtr4i 2208	1 |- { x e. A | ph } = { x e. A | ps }

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   {cab 2163   {crab 2459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-rab 2464

This theorem is referenced by:  rabbii  2723  bm2.5ii  4495  fndmdifcom  5622  cauappcvgprlemladdru  7654  cauappcvgprlemladdrl  7655  cauappcvgpr  7660  caucvgprlemcl  7674  caucvgprlemladdrl  7676  caucvgpr  7680  caucvgprprlemclphr  7703  ioopos  9948  gcdcom  11968  gcdass  12010  lcmcom  12058  lcmass  12079