Theorem vtocldf 2815

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 115	. . 3 |- ( ph -> ( x = A -> ( ps <-> ch ) ) )
6	3, 5	alrimi 1536	. 2 |- ( ph -> A. x ( x = A -> ( ps <-> ch ) ) )
7		vtocld.3	. . 3 |- ( ph -> ps )
8	3, 7	alrimi 1536	. 2 |- ( ph -> A. x ps )
9		vtocld.1	. 2 |- ( ph -> A e. V )
10		vtoclgft 2814	. 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 1260	1 |- ( ph -> ch )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1362   = wceq 1364   F/wnf 1474   e. wcel 2167   F/_wnfc 2326

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765

This theorem is referenced by:  vtocld  2816  peano2  4631  iota2df  5244