Theorem cla4egf 1864

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 1006	. . . 4 |- (-. ps -> A.x -. ps)
4		cla4gf.3	. . . . 5 |- (x = A -> (ph <-> ps))
5	4	negbid 613	. . . 4 |- (x = A -> (-. ph <-> -. ps))
6	1, 3, 5	cla4gf 1863	. . 3 |- (A e. B -> (A.x -. ph -> -. ps))
7	6	con2d 91	. 2 |- (A e. B -> (ps -> -. A.x -. ph))
8		df-ex 983	. 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 956   = wceq 958   e. wcel 960  E.wex 982

This theorem is referenced by:  cla4egv 1866  rcla4e 1875  onminex 3026  zfrep6 3620  tgval3t 7624

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815

Copyright terms: Public domain