Theorem vtoclf 2909

Description: Implicit substitution of a class for a setvar variable. This is a generalization of chvar 1986. (Contributed by NM, 30-Aug-1993.)

Hypotheses

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

Assertion

Ref	Expression
vtoclf	|- ps

Distinct variable group:   x,A

Allowed substitution hints:   ph(x)   ps(x)

Proof of Theorem vtoclf

Step	Hyp	Ref	Expression
1		vtoclf.1	. . 3 |- F/xps
2		vtoclf.2	. . . . 5 |- A e. _V
3	2	isseti 2866	. . . 4 |- E.x x = A
4		vtoclf.3	. . . . . 6 |- (x = A -> (ph <-> ps))
5	4	biimpd 198	. . . . 5 |- (x = A -> (ph -> ps))
6	5	eximi 1576	. . . 4 |- (E.x x = A -> E.x(ph -> ps))
7	3, 6	ax-mp 5	. . 3 |- E.x(ph -> ps)
8	1, 7	19.36i 1872	. 2 |- (A.xph -> ps)
9		vtoclf.4	. 2 |- ph
10	8, 9	mpg 1548	1 |- ps

Syntax hints:   -> wi 4   <-> wb 176  E.wex 1541   F/wnf 1544   = wceq 1642   e. wcel 1710  _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-11 1746  ax-ext 2334

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

This theorem is referenced by:  vtocl  2910