Theorem ralbi12f 33969

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

Hypotheses

Ref	Expression
ralbi12f.1	|- F/_ x A
ralbi12f.2	|- F/_ x B

Assertion

Ref	Expression
ralbi12f	|- ( ( A = B /\ A. x e. A ( ph <-> ps ) ) -> ( A. x e. A ph <-> A. x e. B ps ) )

Proof of Theorem ralbi12f

Step	Hyp	Ref	Expression
1		ralbi 3068	. 2 |- ( A. x e. A ( ph <-> ps ) -> ( A. x e. A ph <-> A. x e. A ps ) )
2		ralbi12f.1	. . 3 |- F/_ x A
3		ralbi12f.2	. . 3 |- F/_ x B
4	2, 3	raleqf 3134	. 2 |- ( A = B -> ( A. x e. A ps <-> A. x e. B ps ) )
5	1, 4	sylan9bbr 737	1 |- ( ( A = B /\ A. x e. A ( ph <-> ps ) ) -> ( A. x e. A ph <-> A. x e. B ps ) )

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

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-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-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917

This theorem is referenced by: (None)