Theorem spc3egv 3198

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

Hypothesis

Ref	Expression
spc3egv.1

Assertion

Ref	Expression
spc3egv

Distinct variable groups:   ,,,   ,,,   ,,,   ,,,

Proof of Theorem spc3egv

Step	Hyp	Ref	Expression
1		elisset 3120	. . . 4
2		elisset 3120	. . . 4
3		elisset 3120	. . . 4
4	1, 2, 3	3anim123i 1181	. . 3
5		eeeanv 1989	. . 3
6	4, 5	sylibr 212	. 2
7		spc3egv.1	. . . . 5
8	7	biimprcd 225	. . . 4
9	8	eximdv 1710	. . 3
10	9	2eximdv 1712	. 2
11	6, 10	syl5com 30	1

Syntax hints:  ->wi 4  <->wb 184  /\w3a 973  =wceq 1395  E.wex 1612  e.wcel 1818

This theorem is referenced by:  spc3gv  3199  dihjatcclem4  37148

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-v 3111