Metamath Proof Explorer


Theorem reubiia

Description: Formula-building rule for restricted existential quantifier (inference form). (Contributed by NM, 14-Nov-2004)

Ref Expression
Hypothesis reubiia.1
|- ( x e. A -> ( ph <-> ps ) )
Assertion reubiia
|- ( E! x e. A ph <-> E! x e. A ps )

Proof

Step Hyp Ref Expression
1 reubiia.1
 |-  ( x e. A -> ( ph <-> ps ) )
2 1 pm5.32i
 |-  ( ( x e. A /\ ph ) <-> ( x e. A /\ ps ) )
3 2 eubii
 |-  ( E! x ( x e. A /\ ph ) <-> E! x ( x e. A /\ ps ) )
4 df-reu
 |-  ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
5 df-reu
 |-  ( E! x e. A ps <-> E! x ( x e. A /\ ps ) )
6 3 4 5 3bitr4i
 |-  ( E! x e. A ph <-> E! x e. A ps )