Theorem rabeqf 3190

Description: Equality theorem for restricted class abstractions, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.)

Hypotheses

Ref	Expression
rabeqf.1	|- F/_ x A
rabeqf.2	|- F/_ x B

Assertion

Ref	Expression
rabeqf	|- ( A = B -> { x e. A | ph } = { x e. B | ph } )

Proof of Theorem rabeqf

Step	Hyp	Ref	Expression
1		rabeqf.1	. . . 4 |- F/_ x A
2		rabeqf.2	. . . 4 |- F/_ x B
3	1, 2	nfeq 2776	. . 3 |- F/ x A = B
4		eleq2 2690	. . . 4 |- ( A = B -> ( x e. A <-> x e. B ) )
5	4	anbi1d 741	. . 3 |- ( A = B -> ( ( x e. A /\ ph ) <-> ( x e. B /\ ph ) ) )
6	3, 5	abbid 2740	. 2 |- ( A = B -> { x | ( x e. A /\ ph ) } = { x | ( x e. B /\ ph ) } )
7		df-rab 2921	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
8		df-rab 2921	. 2 |- { x e. B | ph } = { x | ( x e. B /\ ph ) }
9	6, 7, 8	3eqtr4g 2681	1 |- ( A = B -> { x e. A | ph } = { x e. B | ph } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   F/_wnfc 2751   {crab 2916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921

This theorem is referenced by:  rabeqif  3191  rabeq  3192  fpwrelmapffs  29509  rabeq12f  33965  issmfdf  40946  smfpimltmpt  40955  smfpimltxrmpt  40967  smfpimgtmpt  40989  smfpimgtxrmpt  40992  smfsupmpt  41021  smfinfmpt  41025