Theorem cla4gf 1856

Description: Rule of specialization with implicit substitution. Compare Theorem 7.3 of [Quine] p. 44.

Hypotheses

Ref	Expression
cla4gf.1	|- (y e. A -> A.x y e. A)
cla4gf.2	|- (ps -> A.xps)
cla4gf.3	|- (x = A -> (ph <-> ps))

Assertion

Ref	Expression
cla4gf	|- (A e. B -> (A.xph -> ps))

Distinct variable groups:   x,y   y,A

Proof of Theorem cla4gf

Step	Hyp	Ref	Expression
1		elisset 1813	. 2 |- (A e. B -> A e. V)
2		isset 1810	. . . . 5 |- (A e. V <-> E.y y = A)
3		cla4gf.1	. . . . . . 7 |- (y e. A -> A.x y e. A)
4	3	hbeleq 1564	. . . . . 6 |- (y = A -> A.x y = A)
5		ax-17 969	. . . . . 6 |- (x = A -> A.y x = A)
6		eqeq1 1478	. . . . . 6 |- (y = x -> (y = A <-> x = A))
7	4, 5, 6	cbvex 1164	. . . . 5 |- (E.y y = A <-> E.x x = A)
8	2, 7	bitr 173	. . . 4 |- (A e. V <-> E.x x = A)
9		cla4gf.3	. . . . . 6 |- (x = A -> (ph <-> ps))
10	9	biimpd 153	. . . . 5 |- (x = A -> (ph -> ps))
11	10	19.22i 1038	. . . 4 |- (E.x x = A -> E.x(ph -> ps))
12	8, 11	sylbi 199	. . 3 |- (A e. V -> E.x(ph -> ps))
13		cla4gf.2	. . . 4 |- (ps -> A.xps)
14	13	19.36 1076	. . 3 |- (E.x(ph -> ps) <-> (A.xph -> ps))
15	12, 14	sylib 198	. 2 |- (A e. V -> (A.xph -> ps))
16	1, 15	syl 10	1 |- (A e. B -> (A.xph -> 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:  cla4egf 1857  cla4gv 1858  rcla4 1867  moi 1921

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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