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 )