Theorem rabeqf 2720

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 2320	. . 3 |- F/ x A = B
4		eleq2 2234	. . . 4 |- ( A = B -> ( x e. A <-> x e. B ) )
5	4	anbi1d 462	. . 3 |- ( A = B -> ( ( x e. A /\ ph ) <-> ( x e. B /\ ph ) ) )
6	3, 5	abbid 2287	. 2 |- ( A = B -> { x | ( x e. A /\ ph ) } = { x | ( x e. B /\ ph ) } )
7		df-rab 2457	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
8		df-rab 2457	. 2 |- { x e. B | ph } = { x | ( x e. B /\ ph ) }
9	6, 7, 8	3eqtr4g 2228	1 |- ( A = B -> { x e. A | ph } = { x e. B | ph } )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   {cab 2156   F/_wnfc 2299   {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-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  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-clel 2166  df-nfc 2301  df-rab 2457

This theorem is referenced by:  rabeqif  2721  rabeq  2722