Theorem cla4gf 2361

Description: Rule of specialization, using implicit substitition. Compare Theorem 7.3 of [Quine] p. 44. (Unnecessary distinct variable restrictions were removed by Andrew Salmon, 12-Aug-2011.)

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:   y,A

Proof of Theorem cla4gf

Step	Hyp	Ref	Expression
1		elisset 2299	. 2 |- (A e. B -> A e. _V)
2		isset 2296	. . . . 5 |- (A e. _V <-> E.z z = A)
3		cla4gf.1	. . . . . . . 8 |- (y e. A -> A.x y e. A)
4	3	hblem 1993	. . . . . . 7 |- (z e. A -> A.x z e. A)
5	4	hbeleq 1997	. . . . . 6 |- (z = A -> A.x z = A)
6		ax-17 1317	. . . . . 6 |- (x = A -> A.z x = A)
7		eqeq1 1890	. . . . . 6 |- (z = x -> (z = A <-> x = A))
8	5, 6, 7	cbvex 1529	. . . . 5 |- (E.z z = A <-> E.x x = A)
9	2, 8	bitri 190	. . . 4 |- (A e. _V <-> E.x x = A)
10		cla4gf.3	. . . . . 6 |- (x = A -> (ph <-> ps))
11	10	biimpd 170	. . . . 5 |- (x = A -> (ph -> ps))
12	11	eximi 1387	. . . 4 |- (E.x x = A -> E.x(ph -> ps))
13	9, 12	sylbi 216	. . 3 |- (A e. _V -> E.x(ph -> ps))
14		cla4gf.2	. . . 4 |- (ps -> A.xps)
15	14	19.36 1429	. . 3 |- (E.x(ph -> ps) <-> (A.xph -> ps))
16	13, 15	sylib 215	. 2 |- (A e. _V -> (A.xph -> ps))
17	1, 16	syl 12	1 |- (A e. B -> (A.xph -> ps))

Syntax hints:   -> wi 3   <-> wb 163  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  _Vcvv 2292

This theorem is referenced by:  cla4egf 2362  cla4gv 2364  rcla4 2373  rcla4OLD 2374  elabgt 2400  moi 2436  ac6sfilem1 5506  ac6sfilem3 5508  ordtypelem5 5688  ordtypelem6 5689  ordtypelem5OLD 15379  ordtypelem6OLD 15380

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294

Copyright terms: Public domain