Theorem spc3gv 2945

Description: Specialization with 3 quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.)

Hypothesis

Ref	Expression
spc3egv.1	|- ((x = A /\ y = B /\ z = C) -> (ph <-> ps))

Assertion

Ref	Expression
spc3gv	|- ((A e. V /\ B e. W /\ C e. X) -> (A.xA.yA.zph -> ps))

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

Allowed substitution hints:   ph(x,y,z)   V(x,y,z)   W(x,y,z)   X(x,y,z)

Proof of Theorem spc3gv

Step	Hyp	Ref	Expression
1		spc3egv.1	. . . . 5 |- ((x = A /\ y = B /\ z = C) -> (ph <-> ps))
2	1	notbid 285	. . . 4 |- ((x = A /\ y = B /\ z = C) -> (-. ph <-> -. ps))
3	2	spc3egv 2944	. . 3 |- ((A e. V /\ B e. W /\ C e. X) -> (-. ps -> E.xE.yE.z -. ph))
4		exnal 1574	. . . . . . 7 |- (E.z -. ph <-> -. A.zph)
5	4	exbii 1582	. . . . . 6 |- (E.yE.z -. ph <-> E.y -. A.zph)
6		exnal 1574	. . . . . 6 |- (E.y -. A.zph <-> -. A.yA.zph)
7	5, 6	bitri 240	. . . . 5 |- (E.yE.z -. ph <-> -. A.yA.zph)
8	7	exbii 1582	. . . 4 |- (E.xE.yE.z -. ph <-> E.x -. A.yA.zph)
9		exnal 1574	. . . 4 |- (E.x -. A.yA.zph <-> -. A.xA.yA.zph)
10	8, 9	bitr2i 241	. . 3 |- (-. A.xA.yA.zph <-> E.xE.yE.z -. ph)
11	3, 10	syl6ibr 218	. 2 |- ((A e. V /\ B e. W /\ C e. X) -> (-. ps -> -. A.xA.yA.zph))
12	11	con4d 97	1 |- ((A e. V /\ B e. W /\ C e. X) -> (A.xA.yA.zph -> ps))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   /\ w3a 934  A.wal 1540  E.wex 1541   = wceq 1642   e. wcel 1710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2862

This theorem is referenced by:  fununiq  5518