Theorem vtocld 3257

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 1843	. 2 |- F/ x ph
5		nfcvd 2765	. 2 |- ( ph -> F/_ x A )
6		nfvd 1844	. 2 |- ( ph -> F/ x ch )
7	1, 2, 3, 4, 5, 6	vtocldf 3256	1 |- ( ph -> ch )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202

This theorem is referenced by:  lmatfval  29880  lmatcl  29882  dvgrat  38511