Theorem reubida 2540

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 440	. . 3 |- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. A /\ ch ) ) )
4	1, 3	eubid 1950	. 2 |- ( ph -> ( E! x ( x e. A /\ ps ) <-> E! x ( x e. A /\ ch ) ) )
5		df-reu 2360	. 2 |- ( E! x e. A ps <-> E! x ( x e. A /\ ps ) )
6		df-reu 2360	. 2 |- ( E! x e. A ch <-> E! x ( x e. A /\ ch ) )
7	4, 5, 6	3bitr4g 221	1 |- ( ph -> ( E! x e. A ps <-> E! x e. A ch ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   F/wnf 1390   e. wcel 1434   E!weu 1943   E!wreu 2355

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-eu 1946  df-reu 2360

This theorem is referenced by:  reubidva  2541