This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem ceqsrexv

Description: Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 30-Apr-2004)

Ref Expression
Hypothesis ceqsrexv.1 
|- ( x = A -> ( ph <-> ps ) )
Assertion ceqsrexv 
|- ( A e. B -> ( E. x e. B ( x = A /\ ph ) <-> ps ) )

Proof

Step Hyp Ref Expression
1 ceqsrexv.1 
 |-  ( x = A -> ( ph <-> ps ) )
2 df-rex 
 |-  ( E. x e. B ( x = A /\ ph ) <-> E. x ( x e. B /\ ( x = A /\ ph ) ) )
3 an12 
 |-  ( ( x = A /\ ( x e. B /\ ph ) ) <-> ( x e. B /\ ( x = A /\ ph ) ) )
4 3 exbii 
 |-  ( E. x ( x = A /\ ( x e. B /\ ph ) ) <-> E. x ( x e. B /\ ( x = A /\ ph ) ) )
5 2 4 bitr4i 
 |-  ( E. x e. B ( x = A /\ ph ) <-> E. x ( x = A /\ ( x e. B /\ ph ) ) )
6 eleq1 
 |-  ( x = A -> ( x e. B <-> A e. B ) )
7 6 1 anbi12d 
 |-  ( x = A -> ( ( x e. B /\ ph ) <-> ( A e. B /\ ps ) ) )
8 7 ceqsexgv 
 |-  ( A e. B -> ( E. x ( x = A /\ ( x e. B /\ ph ) ) <-> ( A e. B /\ ps ) ) )
9 8 bianabs 
 |-  ( A e. B -> ( E. x ( x = A /\ ( x e. B /\ ph ) ) <-> ps ) )
10 5 9 bitrid 
 |-  ( A e. B -> ( E. x e. B ( x = A /\ ph ) <-> ps ) )