Theorem ralbid2 2554

Description: Formula-building rule for restricted universal quantifier (deduction form). (Contributed by BJ, 14-Jul-2024.)

Hypotheses

Ref	Expression
ralbid2.nf	|- F/ x ph
ralbid2.1	|- ( ph -> ( ( x e. A -> ps ) <-> ( x e. B -> ch ) ) )

Assertion

Ref	Expression
ralbid2	|- ( ph -> ( A. x e. A ps <-> A. x e. B ch ) )

Proof of Theorem ralbid2

Step	Hyp	Ref	Expression
1		ralbid2.nf	. . 3 |- F/ x ph
2		ralbid2.1	. . 3 |- ( ph -> ( ( x e. A -> ps ) <-> ( x e. B -> ch ) ) )
3	1, 2	albid 1668	. 2 |- ( ph -> ( A. x ( x e. A -> ps ) <-> A. x ( x e. B -> ch ) ) )
4		df-ral 2533	. 2 |- ( A. x e. A ps <-> A. x ( x e. A -> ps ) )
5		df-ral 2533	. 2 |- ( A. x e. B ch <-> A. x ( x e. B -> ch ) )
6	3, 4, 5	3bitr4g 223	1 |- ( ph -> ( A. x e. A ps <-> A. x e. B ch ) )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   A.wal 1400   F/wnf 1513   e. wcel 2209   A.wral 2528

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-4 1563

This proof depends on definitions:  df-bi 117  df-nf 1514  df-ral 2533

This theorem is used by:  strcollnft  17010