Theorem raleqd 39325

Description: Equality deduction for restricted universal quantifier. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
raleqd.a	|- F/_ x A
raleqd.b	|- F/_ x B
raleqd.e	|- ( ph -> A = B )

Assertion

Ref	Expression
raleqd	|- ( ph -> ( A. x e. A ps <-> A. x e. B ps ) )

Proof of Theorem raleqd

Step	Hyp	Ref	Expression
1		raleqd.e	. 2 |- ( ph -> A = B )
2		raleqd.a	. . 3 |- F/_ x A
3		raleqd.b	. . 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	syl 17	1 |- ( ph -> ( A. x e. A ps <-> A. x e. B ps ) )

Syntax hints:   -> wi 4   <-> wb 196   = 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:  allbutfiinf  39647