Theorem vtocld

Description: Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016) Avoid ax-10 , ax-11 , ax-12 . (Revised by SN, 2-Sep-2024)

	Ref	Expression
Hypotheses	vtocld.1	|- ( ph -> A e. V )
	vtocld.2	|- ( ( ph /\ x = A ) -> ( ps <-> ch ) )
	vtocld.3	|- ( ph -> ps )
Assertion	vtocld	|- ( ph -> ch )

Proof

Step	Hyp	Ref	Expression
1		vtocld.1	|- ( ph -> A e. V )
2		vtocld.2	|- ( ( ph /\ x = A ) -> ( ps <-> ch ) )
3		vtocld.3	|- ( ph -> ps )
4		elisset	|- ( A e. V -> E. x x = A )
5	1 4	syl	|- ( ph -> E. x x = A )
6	3	adantr	|- ( ( ph /\ x = A ) -> ps )
7	6 2	mpbid	|- ( ( ph /\ x = A ) -> ch )
8	5 7	exlimddv	|- ( ph -> ch )