Theorem cla42gv 1861

Description: Specialization with 2 quantifiers, using implicit substitution.

Hypothesis

Ref	Expression
cla42egv.1	|- ((x = A /\ y = B) -> (ph <-> ps))

Assertion

Ref	Expression
cla42gv	|- ((A e. C /\ B e. D) -> (A.xA.yph -> ps))

Distinct variable groups:   x,y,A   x,B,y   ps,x,y

Proof of Theorem cla42gv

Step	Hyp	Ref	Expression
1		cla42egv.1	. . . . 5 |- ((x = A /\ y = B) -> (ph <-> ps))
2	1	negbid 610	. . . 4 |- ((x = A /\ y = B) -> (-. ph <-> -. ps))
3	2	cla42egv 1860	. . 3 |- ((A e. C /\ B e. D) -> (-. ps -> E.xE.y -. ph))
4		exnal 1036	. . . . 5 |- (E.y -. ph <-> -. A.yph)
5	4	exbii 1049	. . . 4 |- (E.xE.y -. ph <-> E.x -. A.yph)
6		exnal 1036	. . . 4 |- (E.x -. A.yph <-> -. A.xA.yph)
7	5, 6	bitr2 174	. . 3 |- (-. A.xA.yph <-> E.xE.y -. ph)
8	3, 7	syl6ibr 213	. 2 |- ((A e. C /\ B e. D) -> (-. ps -> -. A.xA.yph))
9	8	a3d 75	1 |- ((A e. C /\ B e. D) -> (A.xA.yph -> ps))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223  A.wal 952   = wceq 954   e. wcel 956  E.wex 978

This theorem is referenced by:  pslem 8590

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808

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