Theorem vtoclgaf 1848

Description: Implicit substitution of a class for a set variable.

Hypotheses

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

Assertion

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

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

Proof of Theorem vtoclgaf

Step	Hyp	Ref	Expression
1		vtoclgaf.1	. . 3 |- (y e. A -> A.x y e. A)
2		ax-17 970	. . . . 5 |- (y e. B -> A.x y e. B)
3	1, 2	hbel 1564	. . . 4 |- (A e. B -> A.x A e. B)
4		vtoclgaf.2	. . . 4 |- (ps -> A.xps)
5	3, 4	hbim 1006	. . 3 |- ((A e. B -> ps) -> A.x(A e. B -> ps))
6		eleq1 1532	. . . 4 |- (x = A -> (x e. B <-> A e. B))
7		vtoclgaf.3	. . . 4 |- (x = A -> (ph <-> ps))
8	6, 7	imbi12d 625	. . 3 |- (x = A -> ((x e. B -> ph) <-> (A e. B -> ps)))
9		vtoclgaf.4	. . 3 |- (x e. B -> ph)
10	1, 5, 8, 9	vtoclgf 1843	. 2 |- (A e. B -> (A e. B -> ps))
11	10	pm2.43i 64	1 |- (A e. B -> ps)

Syntax hints:   -> wi 3   <-> wb 146  A.wal 953   = wceq 955   e. wcel 957

This theorem is referenced by:  vtoclga 1849  isumnn0nna 7160  cnlnadjlem5 9960

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-v 1809

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