Theorem vtocld 2830

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 1552	. 2 |- F/ x ph
5		nfcvd 2351	. 2 |- ( ph -> F/_ x A )
6		nfvd 1553	. 2 |- ( ph -> F/ x ch )
7	1, 2, 3, 4, 5, 6	vtocldf 2829	1 |- ( ph -> ch )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2178

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778

This theorem is referenced by:  funfvima3  5841  isbth  7095  frec2uzuzd  10584  setscomd  12988