Theorem vtoclgf 2914

Description: Implicit substitution of a class for a setvar variable, with bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.) (Proof shortened by Mario Carneiro, 10-Oct-2016.)

Hypotheses

Ref	Expression
vtoclgf.1	|- F/_xA
vtoclgf.2	|- F/xps
vtoclgf.3	|- (x = A -> (ph <-> ps))
vtoclgf.4	|- ph

Assertion

Ref	Expression
vtoclgf	|- (A e. V -> ps)

Proof of Theorem vtoclgf

Step	Hyp	Ref	Expression
1		elex 2868	. 2 |- (A e. V -> A e. _V)
2		vtoclgf.1	. . . 4 |- F/_xA
3	2	issetf 2865	. . 3 |- (A e. _V <-> E.x x = A)
4		vtoclgf.2	. . . 4 |- F/xps
5		vtoclgf.4	. . . . 5 |- ph
6		vtoclgf.3	. . . . 5 |- (x = A -> (ph <-> ps))
7	5, 6	mpbii 202	. . . 4 |- (x = A -> ps)
8	4, 7	exlimi 1803	. . 3 |- (E.x x = A -> ps)
9	3, 8	sylbi 187	. 2 |- (A e. _V -> ps)
10	1, 9	syl 15	1 |- (A e. V -> ps)

Syntax hints:   -> wi 4   <-> wb 176  E.wex 1541   F/wnf 1544   = wceq 1642   e. wcel 1710   F/_wnfc 2477  _Vcvv 2860

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-or 359  df-an 360  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:  vtoclg  2915  vtocl2gf  2917  vtocl3gf  2918  vtoclgaf  2920  ceqsexg  2971  elabgf  2984  mob  3019  opeliunxp2  4823