Theorem cla43gv 1865

Description: Specialization with 3 quantifiers, using implicit substitution.

Hypothesis

Ref	Expression
cla43egv.1	|- ((x = A /\ y = B /\ z = C) -> (ph <-> ps))

Assertion

Ref	Expression
cla43gv	|- ((A e. R /\ B e. S /\ C e. T) -> (A.xA.yA.zph -> ps))

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

Proof of Theorem cla43gv

Step	Hyp	Ref	Expression
1		cla43egv.1	. . . . 5 |- ((x = A /\ y = B /\ z = C) -> (ph <-> ps))
2	1	negbid 610	. . . 4 |- ((x = A /\ y = B /\ z = C) -> (-. ph <-> -. ps))
3	2	cla43egv 1864	. . 3 |- ((A e. R /\ B e. S /\ C e. T) -> (-. ps -> E.xE.yE.z -. ph))
4		exnal 1037	. . . . . . 7 |- (E.z -. ph <-> -. A.zph)
5	4	exbii 1050	. . . . . 6 |- (E.yE.z -. ph <-> E.y -. A.zph)
6		exnal 1037	. . . . . 6 |- (E.y -. A.zph <-> -. A.yA.zph)
7	5, 6	bitr 173	. . . . 5 |- (E.yE.z -. ph <-> -. A.yA.zph)
8	7	exbii 1050	. . . 4 |- (E.xE.yE.z -. ph <-> E.x -. A.yA.zph)
9		exnal 1037	. . . 4 |- (E.x -. A.yA.zph <-> -. A.xA.yA.zph)
10	8, 9	bitr2 174	. . 3 |- (-. A.xA.yA.zph <-> E.xE.yE.z -. ph)
11	3, 10	syl6ibr 213	. 2 |- ((A e. R /\ B e. S /\ C e. T) -> (-. ps -> -. A.xA.yA.zph))
12	11	a3d 75	1 |- ((A e. R /\ B e. S /\ C e. T) -> (A.xA.yA.zph -> ps))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ w3a 774  A.wal 953   = wceq 955   e. wcel 957  E.wex 979

This theorem is referenced by:  funopg 3544  pslem 8630

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1810

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