Theorem vtocl2 2911

Description: Implicit substitution of classes for setvar variables. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Hypotheses

Ref	Expression
vtocl2.1	|- A e. _V
vtocl2.2	|- B e. _V
vtocl2.3	|- ((x = A /\ y = B) -> (ph <-> ps))
vtocl2.4	|- ph

Assertion

Ref	Expression
vtocl2	|- ps

Distinct variable groups:   x,y,A   x,B,y   ps,x,y

Allowed substitution hints:   ph(x,y)

Proof of Theorem vtocl2

Step	Hyp	Ref	Expression
1		vtocl2.1	. . . . . 6 |- A e. _V
2	1	isseti 2866	. . . . 5 |- E.x x = A
3		vtocl2.2	. . . . . 6 |- B e. _V
4	3	isseti 2866	. . . . 5 |- E.y y = B
5		eeanv 1913	. . . . . 6 |- (E.xE.y(x = A /\ y = B) <-> (E.x x = A /\ E.y y = B))
6		vtocl2.3	. . . . . . . 8 |- ((x = A /\ y = B) -> (ph <-> ps))
7	6	biimpd 198	. . . . . . 7 |- ((x = A /\ y = B) -> (ph -> ps))
8	7	2eximi 1577	. . . . . 6 |- (E.xE.y(x = A /\ y = B) -> E.xE.y(ph -> ps))
9	5, 8	sylbir 204	. . . . 5 |- ((E.x x = A /\ E.y y = B) -> E.xE.y(ph -> ps))
10	2, 4, 9	mp2an 653	. . . 4 |- E.xE.y(ph -> ps)
11		19.36v 1896	. . . . 5 |- (E.y(ph -> ps) <-> (A.yph -> ps))
12	11	exbii 1582	. . . 4 |- (E.xE.y(ph -> ps) <-> E.x(A.yph -> ps))
13	10, 12	mpbi 199	. . 3 |- E.x(A.yph -> ps)
14	13	19.36aiv 1897	. 2 |- (A.xA.yph -> ps)
15		vtocl2.4	. . 3 |- ph
16	15	ax-gen 1546	. 2 |- A.yph
17	14, 16	mpg 1548	1 |- ps

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541   = 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-7 1734  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:  caovord  5630