Theorem cla4egf 2362

Description: Existential specialization, using implicit substitition.

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
cla4egf	|- (A e. B -> (ps -> E.xph))

Distinct variable group:   y,A

Proof of Theorem cla4egf

Step	Hyp	Ref	Expression
1		cla4gf.1	. . . 4 |- (y e. A -> A.x y e. A)
2		cla4gf.2	. . . . 5 |- (ps -> A.xps)
3	2	hbn 1351	. . . 4 |- (-. ps -> A.x -. ps)
4		cla4gf.3	. . . . 5 |- (x = A -> (ph <-> ps))
5	4	notbid 673	. . . 4 |- (x = A -> (-. ph <-> -. ps))
6	1, 3, 5	cla4gf 2361	. . 3 |- (A e. B -> (A.x -. ph -> -. ps))
7	6	con2d 107	. 2 |- (A e. B -> (ps -> -. A.x -. ph))
8		df-ex 1327	. 2 |- (E.xph <-> -. A.x -. ph)
9	7, 8	syl6ibr 230	1 |- (A e. B -> (ps -> E.xph))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326

This theorem is referenced by:  cla4egv 2365  rcla4e 2375  onminex 3888  zfrep6 4545  ac6sfi 5509  tgval3 8895  oprabopabf 10157

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

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