Theorem rabeqf 2728

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 2327	. . 3 |- F/ x A = B
4		eleq2 2241	. . . 4 |- ( A = B -> ( x e. A <-> x e. B ) )
5	4	anbi1d 465	. . 3 |- ( A = B -> ( ( x e. A /\ ph ) <-> ( x e. B /\ ph ) ) )
6	3, 5	abbid 2294	. 2 |- ( A = B -> { x | ( x e. A /\ ph ) } = { x | ( x e. B /\ ph ) } )
7		df-rab 2464	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
8		df-rab 2464	. 2 |- { x e. B | ph } = { x | ( x e. B /\ ph ) }
9	6, 7, 8	3eqtr4g 2235	1 |- ( A = B -> { x e. A | ph } = { x e. B | ph } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   {cab 2163   F/_wnfc 2306   {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-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  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-clel 2173  df-nfc 2308  df-rab 2464

This theorem is referenced by:  rabeqif  2729  rabeq  2730