Theorem reubida 3124

Description: Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by Mario Carneiro, 19-Nov-2016.)

Hypotheses

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

Assertion

Ref	Expression
reubida	|- ( ph -> ( E! x e. A ps <-> E! x e. A ch ) )

Proof of Theorem reubida

Step	Hyp	Ref	Expression
1		reubida.1	. . 3 |- F/ x ph
2		reubida.2	. . . 4 |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
3	2	pm5.32da 673	. . 3 |- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. A /\ ch ) ) )
4	1, 3	eubid 2488	. 2 |- ( ph -> ( E! x ( x e. A /\ ps ) <-> E! x ( x e. A /\ ch ) ) )
5		df-reu 2919	. 2 |- ( E! x e. A ps <-> E! x ( x e. A /\ ps ) )
6		df-reu 2919	. 2 |- ( E! x e. A ch <-> E! x ( x e. A /\ ch ) )
7	4, 5, 6	3bitr4g 303	1 |- ( ph -> ( E! x e. A ps <-> E! x e. A ch ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   F/wnf 1708   e. wcel 1990   E!weu 2470   E!wreu 2914

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-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-nf 1710  df-eu 2474  df-reu 2919

This theorem is referenced by:  reubidva  3125  poimirlem25  33434  reuan  41180