Theorem rabeq12f 33965

Description: Equality deduction for restricted class abstraction. (Contributed by Giovanni Mascellani, 10-Apr-2018.)

Hypotheses

Ref	Expression
rabeq12f.1	|- F/_ x A
rabeq12f.2	|- F/_ x B

Assertion

Ref	Expression
rabeq12f	|- ( ( A = B /\ A. x e. A ( ph <-> ps ) ) -> { x e. A | ph } = { x e. B | ps } )

Proof of Theorem rabeq12f

Step	Hyp	Ref	Expression
1		rabbi 3120	. . 3 |- ( A. x e. A ( ph <-> ps ) <-> { x e. A | ph } = { x e. A | ps } )
2	1	biimpi 206	. 2 |- ( A. x e. A ( ph <-> ps ) -> { x e. A | ph } = { x e. A | ps } )
3		rabeq12f.1	. . 3 |- F/_ x A
4		rabeq12f.2	. . 3 |- F/_ x B
5	3, 4	rabeqf 3190	. 2 |- ( A = B -> { x e. A | ps } = { x e. B | ps } )
6	2, 5	sylan9eqr 2678	1 |- ( ( A = B /\ A. x e. A ( ph <-> ps ) ) -> { x e. A | ph } = { x e. B | ps } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   F/_wnfc 2751   A.wral 2912   {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-ral 2917  df-rab 2921

This theorem is referenced by: (None)