Theorem spc3egv 2943

Description: Existential 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
spc3egv	|- ((A e. V /\ B e. W /\ C e. X) -> (ps -> E.xE.yE.zph))

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 spc3egv

Step	Hyp	Ref	Expression
1		elisset 2869	. . . 4 |- (A e. V -> E.x x = A)
2		elisset 2869	. . . 4 |- (B e. W -> E.y y = B)
3		elisset 2869	. . . 4 |- (C e. X -> E.z z = C)
4	1, 2, 3	3anim123i 1137	. . 3 |- ((A e. V /\ B e. W /\ C e. X) -> (E.x x = A /\ E.y y = B /\ E.z z = C))
5		eeeanv 1914	. . 3 |- (E.xE.yE.z(x = A /\ y = B /\ z = C) <-> (E.x x = A /\ E.y y = B /\ E.z z = C))
6	4, 5	sylibr 203	. 2 |- ((A e. V /\ B e. W /\ C e. X) -> E.xE.yE.z(x = A /\ y = B /\ z = C))
7		spc3egv.1	. . . . 5 |- ((x = A /\ y = B /\ z = C) -> (ph <-> ps))
8	7	biimprcd 216	. . . 4 |- (ps -> ((x = A /\ y = B /\ z = C) -> ph))
9	8	eximdv 1622	. . 3 |- (ps -> (E.z(x = A /\ y = B /\ z = C) -> E.zph))
10	9	2eximdv 1624	. 2 |- (ps -> (E.xE.yE.z(x = A /\ y = B /\ z = C) -> E.xE.yE.zph))
11	6, 10	syl5com 26	1 |- ((A e. V /\ B e. W /\ C e. X) -> (ps -> E.xE.yE.zph))

Syntax hints:   -> wi 4   <-> wb 176   /\ w3a 934  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 2861

This theorem is referenced by:  spc3gv  2944