Theorem vtocl2 1379

Description: Implicit substitution of classes for set variables.

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 group(s):   x,y,A   x,B,y   ps,x,y

Proof of Theorem vtocl2

Step	Hyp	Ref	Expression
1		vtocl2.1	. . . . 5 |- A e. V
2	1	isseti 1352	. . . 4 |- E.x x = A
3		vtocl2.2	. . . . 5 |- B e. V
4	3	isseti 1352	. . . 4 |- E.y y = B
5		eeanv 980	. . . . 5 |- (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 135	. . . . . . 7 |- ((x = A /\ y = B) -> (ph -> ps))
8	7	19.22i 723	. . . . . 6 |- (E.y(x = A /\ y = B) -> E.y(ph -> ps))
9	8	19.22i 723	. . . . 5 |- (E.xE.y(x = A /\ y = B) -> E.xE.y(ph -> ps))
10	5, 9	sylbir 176	. . . 4 |- ((E.x x = A /\ E.y y = B) -> E.xE.y(ph -> ps))
11	2, 4, 10	mp2an 520	. . 3 |- E.xE.y(ph -> ps)
12		19.36v 958	. . . . 5 |- (E.y(ph -> ps) <-> (A.yph -> ps))
13	12	biex 733	. . . 4 |- (E.xE.y(ph -> ps) <-> E.x(A.yph -> ps))
14		19.36v 958	. . . 4 |- (E.x(A.yph -> ps) <-> (A.xA.yph -> ps))
15	13, 14	bitr 151	. . 3 |- (E.xE.y(ph -> ps) <-> (A.xA.yph -> ps))
16	11, 15	mpbi 164	. 2 |- (A.xA.yph -> ps)
17		vtocl2.4	. . 3 |- ph
18	17	ax-gen 677	. 2 |- A.yph
19	16, 18	mpg 684	1 |- ps

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678   = wceq 1091   e. wcel 1092  Vcvv 1348

This theorem is referenced by:  caoprcom 3067  caoprord 3070  ersym 3209

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-9 799  ax-12 802  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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