Theorem vtocld 2782

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

Hypotheses

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

Assertion

Ref	Expression
vtocld	|- ( ph -> ch )

Distinct variable groups:   x, A   ph, x   ch, x

Allowed substitution hints:   ps( x)   V( x)

Proof of Theorem vtocld

Step	Hyp	Ref	Expression
1		vtocld.1	. 2 |- ( ph -> A e. V )
2		vtocld.2	. 2 |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )
3		vtocld.3	. 2 |- ( ph -> ps )
4		nfv 1521	. 2 |- F/ x ph
5		nfcvd 2313	. 2 |- ( ph -> F/_ x A )
6		nfvd 1522	. 2 |- ( ph -> F/ x ch )
7	1, 2, 3, 4, 5, 6	vtocldf 2781	1 |- ( ph -> ch )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732

This theorem is referenced by:  funfvima3  5729  isbth  6944  frec2uzuzd  10358