Theorem spc2gv 2943

Description: Specialization with 2 quantifiers, using implicit substitution. (Contributed by NM, 27-Apr-2004.)

Hypothesis

Ref	Expression
spc2egv.1	|- ((x = A /\ y = B) -> (ph <-> ps))

Assertion

Ref	Expression
spc2gv	|- ((A e. V /\ B e. W) -> (A.xA.yph -> ps))

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

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

Proof of Theorem spc2gv

Step	Hyp	Ref	Expression
1		spc2egv.1	. . . . 5 |- ((x = A /\ y = B) -> (ph <-> ps))
2	1	notbid 285	. . . 4 |- ((x = A /\ y = B) -> (-. ph <-> -. ps))
3	2	spc2egv 2942	. . 3 |- ((A e. V /\ B e. W) -> (-. ps -> E.xE.y -. ph))
4		2nalexn 1573	. . 3 |- (-. A.xA.yph <-> E.xE.y -. ph)
5	3, 4	syl6ibr 218	. 2 |- ((A e. V /\ B e. W) -> (-. ps -> -. A.xA.yph))
6	5	con4d 97	1 |- ((A e. V /\ B e. W) -> (A.xA.yph -> ps))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   /\ wa 358  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-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:  nnsucelr  4429  ssfin  4471  ncfinlower  4484