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Metamath Proof Explorer

Theorem ceqex

Description: Equality implies equivalence with substitution. (Contributed by NM, 2-Mar-1995) (Proof shortened by BJ, 1-May-2019)

Ref Expression
Assertion ceqex 
|- ( x = A -> ( ph <-> E. x ( x = A /\ ph ) ) )

Proof

Step Hyp Ref Expression
1 19.8a 
 |-  ( ( x = A /\ ph ) -> E. x ( x = A /\ ph ) )
2 1 ex 
 |-  ( x = A -> ( ph -> E. x ( x = A /\ ph ) ) )
3 eqvisset 
 |-  ( x = A -> A e. _V )
4 alexeqg 
 |-  ( A e. _V -> ( A. x ( x = A -> ph ) <-> E. x ( x = A /\ ph ) ) )
5 3 4 syl 
 |-  ( x = A -> ( A. x ( x = A -> ph ) <-> E. x ( x = A /\ ph ) ) )
6 sp 
 |-  ( A. x ( x = A -> ph ) -> ( x = A -> ph ) )
7 6 com12 
 |-  ( x = A -> ( A. x ( x = A -> ph ) -> ph ) )
8 5 7 sylbird 
 |-  ( x = A -> ( E. x ( x = A /\ ph ) -> ph ) )
9 2 8 impbid 
 |-  ( x = A -> ( ph <-> E. x ( x = A /\ ph ) ) )