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Metamath Proof Explorer

Theorem sbcex2

Description: Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004) (Revised by NM, 18-Aug-2018)

Ref Expression
Assertion sbcex2 
|- ( [. A / y ]. E. x ph <-> E. x [. A / y ]. ph )

Proof

Step Hyp Ref Expression
1 sbcex 
 |-  ( [. A / y ]. E. x ph -> A e. _V )
2 sbcex 
 |-  ( [. A / y ]. ph -> A e. _V )
3 2 exlimiv 
 |-  ( E. x [. A / y ]. ph -> A e. _V )
4 dfsbcq2 
 |-  ( z = A -> ( [ z / y ] E. x ph <-> [. A / y ]. E. x ph ) )
5 dfsbcq2 
 |-  ( z = A -> ( [ z / y ] ph <-> [. A / y ]. ph ) )
6 5 exbidv 
 |-  ( z = A -> ( E. x [ z / y ] ph <-> E. x [. A / y ]. ph ) )
7 sbex 
 |-  ( [ z / y ] E. x ph <-> E. x [ z / y ] ph )
8 4 6 7 vtoclbg 
 |-  ( A e. _V -> ( [. A / y ]. E. x ph <-> E. x [. A / y ]. ph ) )
9 1 3 8 pm5.21nii 
 |-  ( [. A / y ]. E. x ph <-> E. x [. A / y ]. ph )