Theorem rmobida 2615

Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 16-Jun-2017.)

Hypotheses

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

Assertion

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

Proof of Theorem rmobida

Step	Hyp	Ref	Expression
1		rmobida.1	. . 3 |- F/ x ph
2		rmobida.2	. . . 4 |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
3	2	pm5.32da 447	. . 3 |- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. A /\ ch ) ) )
4	1, 3	mobid 2032	. 2 |- ( ph -> ( E* x ( x e. A /\ ps ) <-> E* x ( x e. A /\ ch ) ) )
5		df-rmo 2422	. 2 |- ( E* x e. A ps <-> E* x ( x e. A /\ ps ) )
6		df-rmo 2422	. 2 |- ( E* x e. A ch <-> E* x ( x e. A /\ ch ) )
7	4, 5, 6	3bitr4g 222	1 |- ( ph -> ( E* x e. A ps <-> E* x e. A ch ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   F/wnf 1436   e. wcel 1480   E*wmo 1998   E*wrmo 2417

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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-eu 2000  df-mo 2001  df-rmo 2422

This theorem is referenced by:  rmobidva  2616