Theorem reubida

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

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

Proof

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