Theorem ralbid2 2470

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 1603	. 2 |- ( ph -> ( A. x ( x e. A -> ps ) <-> A. x ( x e. B -> ch ) ) )
4		df-ral 2449	. 2 |- ( A. x e. A ps <-> A. x ( x e. A -> ps ) )
5		df-ral 2449	. 2 |- ( A. x e. B ch <-> A. x ( x e. B -> ch ) )
6	3, 4, 5	3bitr4g 222	1 |- ( ph -> ( A. x e. A ps <-> A. x e. B ch ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1341   F/wnf 1448   e. wcel 2136   A.wral 2444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-4 1498

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-ral 2449

This theorem is referenced by:  strcollnft  13866