Theorem cla4e2gv 1398

Description: Existential specialization with 2 quantifiers, using implicit substitution.

Hypothesis

Ref	Expression
cla4e2gv.1	|- ((x = A /\ y = B) -> (ph <-> ps))

Assertion

Ref	Expression
cla4e2gv	|- ((A e. C /\ B e. D) -> (ps -> E.xE.yph))

Distinct variable group(s):   x,y,A   x,B,y   ps,x,y

Proof of Theorem cla4e2gv

Step	Hyp	Ref	Expression
1		cla4e2gv.1	. . . . 5 |- ((x = A /\ y = B) -> (ph <-> ps))
2	1	biimprcd 138	. . . 4 |- (ps -> ((x = A /\ y = B) -> ph))
3	2	19.22dvv 949	. . 3 |- (ps -> (E.xE.y(x = A /\ y = B) -> E.xE.yph))
4		elex 1356	. . . . 5 |- (A e. C -> E.x x = A)
5		elex 1356	. . . . 5 |- (B e. D -> E.y y = B)
6	4, 5	anim12i 268	. . . 4 |- ((A e. C /\ B e. D) -> (E.x x = A /\ E.y y = B))
7		eeanv 980	. . . 4 |- (E.xE.y(x = A /\ y = B) <-> (E.x x = A /\ E.y y = B))
8	6, 7	sylibr 175	. . 3 |- ((A e. C /\ B e. D) -> E.xE.y(x = A /\ y = B))
9	3, 8	syl5 22	. 2 |- (ps -> ((A e. C /\ B e. D) -> E.xE.yph))
10	9	com12 13	1 |- ((A e. C /\ B e. D) -> (ps -> E.xE.yph))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  E.wex 678   = wceq 1091   e. wcel 1092

This theorem is referenced by:  cla42gv 1399  cla4e2v 1406  th3q 3253  genpprecl 3898

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-9 799  ax-12 802  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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