Theorem cla4egf 1852

Description: Existential specialization with implicit substitution.

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 groups:   x,y   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 1001	. . . 4 |- (-. ps -> A.x -. ps)
4		cla4gf.3	. . . . 5 |- (x = A -> (ph <-> ps))
5	4	negbid 609	. . . 4 |- (x = A -> (-. ph <-> -. ps))
6	1, 3, 5	cla4gf 1851	. . 3 |- (A e. B -> (A.x -. ph -> -. ps))
7	6	con2d 91	. 2 |- (A e. B -> (ps -> -. A.x -. ph))
8		df-ex 978	. 2 |- (E.xph <-> -. A.x -. ph)
9	7, 8	syl6ibr 213	1 |- (A e. B -> (ps -> E.xph))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146  A.wal 951   = wceq 953   e. wcel 955  E.wex 977

This theorem is referenced by:  cla4egv 1854  rcla4e 1863  onminex 3010  zfrep6 3600  tgval3t 7567

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803

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