Theorem reubida 2794

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

Hypotheses

Ref	Expression
reubida.1	|- F/xph
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/xph
2		reubida.2	. . . 4 |- ((ph /\ x e. A) -> (ps <-> ch))
3	2	pm5.32da 622	. . 3 |- (ph -> ((x e. A /\ ps) <-> (x e. A /\ ch)))
4	1, 3	eubid 2211	. 2 |- (ph -> (E!x(x e. A /\ ps) <-> E!x(x e. A /\ ch)))
5		df-reu 2622	. 2 |- (E!x e. A ps <-> E!x(x e. A /\ ps))
6		df-reu 2622	. 2 |- (E!x e. A ch <-> E!x(x e. A /\ ch))
7	4, 5, 6	3bitr4g 279	1 |- (ph -> (E!x e. A ps <-> E!x e. A ch))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   F/wnf 1544   e. wcel 1710  E!weu 2204  E!wreu 2617

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-eu 2208  df-reu 2622

This theorem is referenced by:  reubidva  2795