Theorem vtoclgf 1842

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 groups:   y,A   x,y

Proof of Theorem vtoclgf

Step	Hyp	Ref	Expression
1		elisset 1813	. 2 |- (A e. B -> A e. V)
2		isset 1810	. . . 4 |- (A e. V <-> E.y y = A)
3		vtoclgf.1	. . . . . 6 |- (y e. A -> A.x y e. A)
4	3	hbeleq 1564	. . . . 5 |- (y = A -> A.x y = A)
5		ax-17 969	. . . . 5 |- (x = A -> A.y x = A)
6		eqeq1 1478	. . . . 5 |- (y = x -> (y = A <-> x = A))
7	4, 5, 6	cbvex 1164	. . . 4 |- (E.y y = A <-> E.x x = A)
8	2, 7	bitr 173	. . 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 193	. . . 4 |- (x = A -> ps)
13	9, 12	19.23ai 1062	. . 3 |- (E.x x = A -> ps)
14	8, 13	sylbi 199	. 2 |- (A e. V -> ps)
15	1, 14	syl 10	1 |- (A e. B -> ps)

Syntax hints:   -> wi 3   <-> wb 146  A.wal 952   = wceq 954   e. wcel 956  E.wex 978  Vcvv 1807

This theorem is referenced by:  vtoclg 1843  vtocl2gf 1845  vtoclgaf 1847  ceqsexg 1883  elabgf 1894  ssiun2s 2589  reuuni2f 2878

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808

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