Theorem vtocldf 2605

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 )
vtocldf.4	|- F/ x ph
vtocldf.5	|- ( ph -> F/_ x A )
vtocldf.6	|- ( ph -> F/ x ch )

Assertion

Ref	Expression
vtocldf	|- ( ph -> ch )

Proof of Theorem vtocldf

Step	Hyp	Ref	Expression
1		vtocldf.5	. 2 |- ( ph -> F/_ x A )
2		vtocldf.6	. 2 |- ( ph -> F/ x ch )
3		vtocldf.4	. . 3 |- F/ x ph
4		vtocld.2	. . . 4 |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )
5	4	ex 108	. . 3 |- ( ph -> ( x = A -> ( ps <-> ch ) ) )
6	3, 5	alrimi 1415	. 2 |- ( ph -> A. x ( x = A -> ( ps <-> ch ) ) )
7		vtocld.3	. . 3 |- ( ph -> ps )
8	3, 7	alrimi 1415	. 2 |- ( ph -> A. x ps )
9		vtocld.1	. 2 |- ( ph -> A e. V )
10		vtoclgft 2604	. 2 |- ( ( ( F/_ x A /\ F/ x ch ) /\ ( A. x ( x = A -> ( ps <-> ch ) ) /\ A. x ps ) /\ A e. V ) -> ch )
11	1, 2, 6, 8, 9, 10	syl221anc 1146	1 |- ( ph -> ch )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   A.wal 1241   = wceq 1243   F/wnf 1349   e. wcel 1393   F/_wnfc 2165

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559

This theorem is referenced by:  vtocld  2606  peano2  4318  iota2df  4891