Theorem spc2egv 2942

Description: Existential specialization with 2 quantifiers, using implicit substitution. (Contributed by NM, 3-Aug-1995.)

Hypothesis

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

Assertion

Ref	Expression
spc2egv	|- ((A e. V /\ B e. W) -> (ps -> E.xE.yph))

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 spc2egv

Step	Hyp	Ref	Expression
1		elisset 2870	. . . 4 |- (A e. V -> E.x x = A)
2		elisset 2870	. . . 4 |- (B e. W -> E.y y = B)
3	1, 2	anim12i 549	. . 3 |- ((A e. V /\ B e. W) -> (E.x x = A /\ E.y y = B))
4		eeanv 1913	. . 3 |- (E.xE.y(x = A /\ y = B) <-> (E.x x = A /\ E.y y = B))
5	3, 4	sylibr 203	. 2 |- ((A e. V /\ B e. W) -> E.xE.y(x = A /\ y = B))
6		spc2egv.1	. . . 4 |- ((x = A /\ y = B) -> (ph <-> ps))
7	6	biimprcd 216	. . 3 |- (ps -> ((x = A /\ y = B) -> ph))
8	7	2eximdv 1624	. 2 |- (ps -> (E.xE.y(x = A /\ y = B) -> E.xE.yph))
9	5, 8	syl5com 26	1 |- ((A e. V /\ B e. W) -> (ps -> E.xE.yph))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  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:  spc2gv  2943  spc2ev  2948