Theorem vtocldf

Description: Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016)

	Ref	Expression
Hypotheses	vtocld.1	|- ( ph -> A e. V )
	vtocld.2	|- ( ( ph /\ x = A ) -> ( ps <-> ch ) )
	vtocld.3	|- ( ph -> ps )
	vtocldf.4	|- F/ x ph
	vtocldf.5	|- ( ph -> F/_ x A )
	vtocldf.6	|- ( ph -> F/ x ch )
Assertion	vtocldf	|- ( 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		vtocldf.4	|- F/ x ph
5		vtocldf.5	|- ( ph -> F/_ x A )
6		vtocldf.6	|- ( ph -> F/ x ch )
7	2	ex	|- ( ph -> ( x = A -> ( ps <-> ch ) ) )
8	4 7	alrimi	|- ( ph -> A. x ( x = A -> ( ps <-> ch ) ) )
9	4 3	alrimi	|- ( ph -> A. x ps )
10		vtoclgft	|- ( ( ( F/_ x A /\ F/ x ch ) /\ ( A. x ( x = A -> ( ps <-> ch ) ) /\ A. x ps ) /\ A e. V ) -> ch )
11	5 6 8 9 1 10	syl221anc	|- ( ph -> ch )