Theorem vtoclgf 1382

Description: Implicit substitution of a class for a set variable, with bound variable hypotheses in place of distinct variable restrictions.

Hypotheses

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

Assertion

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

Distinct variable group(s):   y,A   x,y

Proof of Theorem vtoclgf

Step	Hyp	Ref	Expression
1		elisset 1354	. 2 |- (A e. B -> A e. V)
2		isset 1351	. . . 4 |- (A e. V <-> E.y y = A)
3		vtoclgf.1	. . . . . 6 |- (y e. A -> A.x y e. A)
4	3	hbeleq 1173	. . . . 5 |- (y = A -> A.x y = A)
5		ax-17 925	. . . . 5 |- (x = A -> A.y x = A)
6		cleq1 1107	. . . . 5 |- (y = x -> (y = A <-> x = A))
7	4, 5, 6	cbvex 849	. . . 4 |- (E.y y = A <-> E.x x = A)
8	2, 7	bitr 151	. . 3 |- (A e. V <-> E.x x = A)
9		vtoclgf.2	. . . 4 |- (ps -> A.xps)
10		vtoclgf.4	. . . . 5 |- ph
11		vtoclgf.3	. . . . 5 |- (x = A -> (ph <-> ps))
12	10, 11	mpbii 168	. . . 4 |- (x = A -> ps)
13	9, 12	19.23ai 746	. . 3 |- (E.x x = A -> ps)
14	8, 13	sylbi 174	. 2 |- (A e. V -> ps)
15	1, 14	syl 12	1 |- (A e. B -> ps)

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

This theorem is referenced by:  vtoclg 1383  vtocl2gf 1385  ceqsexg 1411  elabgf 1416  reuuni2 1956  ssiun2s 2020

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-8 798  ax-9 799  ax-12 802  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  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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