Theorem spc3egv 2822

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. x E. y E. z ph ) )

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 2744	. . . 4 |- ( A e. V -> E. x x = A )
2		elisset 2744	. . . 4 |- ( B e. W -> E. y y = B )
3		elisset 2744	. . . 4 |- ( C e. X -> E. z z = C )
4	1, 2, 3	3anim123i 1179	. . 3 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( E. x x = A /\ E. y y = B /\ E. z z = C ) )
5		eeeanv 1926	. . 3 |- ( E. x E. y E. z ( x = A /\ y = B /\ z = C ) <-> ( E. x x = A /\ E. y y = B /\ E. z z = C ) )
6	4, 5	sylibr 133	. 2 |- ( ( A e. V /\ B e. W /\ C e. X ) -> E. x E. y E. z ( x = A /\ y = B /\ z = C ) )
7		spc3egv.1	. . . . 5 |- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) )
8	7	biimprcd 159	. . . 4 |- ( ps -> ( ( x = A /\ y = B /\ z = C ) -> ph ) )
9	8	eximdv 1873	. . 3 |- ( ps -> ( E. z ( x = A /\ y = B /\ z = C ) -> E. z ph ) )
10	9	2eximdv 1875	. 2 |- ( ps -> ( E. x E. y E. z ( x = A /\ y = B /\ z = C ) -> E. x E. y E. z ph ) )
11	6, 10	syl5com 29	1 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( ps -> E. x E. y E. z ph ) )

Syntax hints:   -> wi 4   <-> wb 104   /\ w3a 973   = wceq 1348   E.wex 1485   e. wcel 2141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-v 2732

This theorem is referenced by: (None)