Theorem cla4gf 1394

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 group(s):   x,y   y,A

Proof of Theorem cla4gf

Step	Hyp	Ref	Expression
1		elisset 1354	. 2 |- (A e. B -> A e. V)
2		isset 1351	. . . . 5 |- (A e. V <-> E.y y = A)
3		cla4gf.1	. . . . . . 7 |- (y e. A -> A.x y e. A)
4	3	hbeleq 1173	. . . . . 6 |- (y = A -> A.x y = A)
5		ax-17 925	. . . . . 6 |- (x = A -> A.y x = A)
6		cleq1 1107	. . . . . 6 |- (y = x -> (y = A <-> x = A))
7	4, 5, 6	cbvex 849	. . . . 5 |- (E.y y = A <-> E.x x = A)
8	2, 7	bitr 151	. . . 4 |- (A e. V <-> E.x x = A)
9		cla4gf.3	. . . . . 6 |- (x = A -> (ph <-> ps))
10	9	biimpd 135	. . . . 5 |- (x = A -> (ph -> ps))
11	10	19.22i 723	. . . 4 |- (E.x x = A -> E.x(ph -> ps))
12	8, 11	sylbi 174	. . 3 |- (A e. V -> E.x(ph -> ps))
13		cla4gf.2	. . . 4 |- (ps -> A.xps)
14	13	19.36 757	. . 3 |- (E.x(ph -> ps) <-> (A.xph -> ps))
15	12, 14	sylib 173	. 2 |- (A e. V -> (A.xph -> ps))
16	1, 15	syl 12	1 |- (A e. B -> (A.xph -> 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:  cla4egf 1395  cla4gv 1396

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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