Theorem rabeqf 2716

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 2316	. . 3 |- F/ x A = B
4		eleq2 2230	. . . 4 |- ( A = B -> ( x e. A <-> x e. B ) )
5	4	anbi1d 461	. . 3 |- ( A = B -> ( ( x e. A /\ ph ) <-> ( x e. B /\ ph ) ) )
6	3, 5	abbid 2283	. 2 |- ( A = B -> { x | ( x e. A /\ ph ) } = { x | ( x e. B /\ ph ) } )
7		df-rab 2453	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
8		df-rab 2453	. 2 |- { x e. B | ph } = { x | ( x e. B /\ ph ) }
9	6, 7, 8	3eqtr4g 2224	1 |- ( A = B -> { x e. A | ph } = { x e. B | ph } )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   {cab 2151   F/_wnfc 2295   {crab 2448

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rab 2453

This theorem is referenced by:  rabeqif  2717  rabeq  2718