Theorem ralbida 2484

Description: Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 6-Oct-2003.)

Hypotheses

Ref	Expression
ralbida.1	|- F/ x ph
ralbida.2	|- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )

Assertion

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

Proof of Theorem ralbida

Step	Hyp	Ref	Expression
1		ralbida.1	. . 3 |- F/ x ph
2		ralbida.2	. . . 4 |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
3	2	pm5.74da 443	. . 3 |- ( ph -> ( ( x e. A -> ps ) <-> ( x e. A -> ch ) ) )
4	1, 3	albid 1626	. 2 |- ( ph -> ( A. x ( x e. A -> ps ) <-> A. x ( x e. A -> ch ) ) )
5		df-ral 2473	. 2 |- ( A. x e. A ps <-> A. x ( x e. A -> ps ) )
6		df-ral 2473	. 2 |- ( A. x e. A ch <-> A. x ( x e. A -> ch ) )
7	4, 5, 6	3bitr4g 223	1 |- ( ph -> ( A. x e. A ps <-> A. x e. A ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1362   F/wnf 1471   e. wcel 2160   A.wral 2468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-4 1521

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-ral 2473

This theorem is referenced by:  ralbidva  2486  ralbid  2488  2ralbida  2511  ralbi  2622  ismkvnex  7183  caucvgsrlemgt1  7824  iswomninnlem  15256