Theorem vtocldf 2907

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/xph
vtocldf.5	|- (ph -> F/_xA)
vtocldf.6	|- (ph -> F/xch)

Assertion

Ref	Expression
vtocldf	|- (ph -> ch)

Proof of Theorem vtocldf

Step	Hyp	Ref	Expression
1		vtocldf.5	. 2 |- (ph -> F/_xA)
2		vtocldf.6	. 2 |- (ph -> F/xch)
3		vtocldf.4	. . 3 |- F/xph
4		vtocld.2	. . . 4 |- ((ph /\ x = A) -> (ps <-> ch))
5	4	ex 423	. . 3 |- (ph -> (x = A -> (ps <-> ch)))
6	3, 5	alrimi 1765	. 2 |- (ph -> A.x(x = A -> (ps <-> ch)))
7		vtocld.3	. . 3 |- (ph -> ps)
8	3, 7	alrimi 1765	. 2 |- (ph -> A.xps)
9		vtocld.1	. 2 |- (ph -> A e. V)
10		vtoclgft 2906	. 2 |- ((( F/_xA /\ F/xch) /\ (A.x(x = A -> (ps <-> ch)) /\ A.xps) /\ A e. V) -> ch)
11	1, 2, 6, 8, 9, 10	syl221anc 1193	1 |- (ph -> ch)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540   F/wnf 1544   = wceq 1642   e. wcel 1710   F/_wnfc 2477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862

This theorem is referenced by:  vtocld  2908