Theorem cla42egv 1864

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

Hypothesis

Ref	Expression
cla42egv.1	|- ((x = A /\ y = B) -> (ph <-> ps))

Assertion

Ref	Expression
cla42egv	|- ((A e. C /\ B e. D) -> (ps -> E.xE.yph))

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

Proof of Theorem cla42egv

Step	Hyp	Ref	Expression
1		cla42egv.1	. . . 4 |- ((x = A /\ y = B) -> (ph <-> ps))
2	1	biimprcd 156	. . 3 |- (ps -> ((x = A /\ y = B) -> ph))
3	2	19.22dvv 1292	. 2 |- (ps -> (E.xE.y(x = A /\ y = B) -> E.xE.yph))
4		elex 1819	. . . 4 |- (A e. C -> E.x x = A)
5		elex 1819	. . . 4 |- (B e. D -> E.y y = B)
6	4, 5	anim12i 333	. . 3 |- ((A e. C /\ B e. D) -> (E.x x = A /\ E.y y = B))
7		eeanv 1323	. . 3 |- (E.xE.y(x = A /\ y = B) <-> (E.x x = A /\ E.y y = B))
8	6, 7	sylibr 200	. 2 |- ((A e. C /\ B e. D) -> E.xE.y(x = A /\ y = B))
9	3, 8	syl5com 52	1 |- ((A e. C /\ B e. D) -> (ps -> E.xE.yph))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980

This theorem is referenced by:  cla42gv 1865  cla42ev 1870  th3q 4317  genpprecl 5104

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812

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