Theorem spc3egv 3297

Description: Existential specialization with three 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 3215	. . . 4 |- ( A e. V -> E. x x = A )
2		elisset 3215	. . . 4 |- ( B e. W -> E. y y = B )
3		elisset 3215	. . . 4 |- ( C e. X -> E. z z = C )
4	1, 2, 3	3anim123i 1247	. . 3 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( E. x x = A /\ E. y y = B /\ E. z z = C ) )
5		eeeanv 2183	. . 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 224	. 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 240	. . . 4 |- ( ps -> ( ( x = A /\ y = B /\ z = C ) -> ph ) )
9	8	eximdv 1846	. . 3 |- ( ps -> ( E. z ( x = A /\ y = B /\ z = C ) -> E. z ph ) )
10	9	2eximdv 1848	. 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 31	1 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( ps -> E. x E. y E. z ph ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202

This theorem is referenced by:  spc3gv  3298  dihjatcclem4  36710