Theorem cla4e2v 1406

Description: Existential specialization with implicit substitution.

Hypotheses

Ref	Expression
cla4e2v.1	|- A e. V
cla4e2v.2	|- B e. V
cla4e2v.3	|- ((x = A /\ y = B) -> (ph <-> ps))

Assertion

Ref	Expression
cla4e2v	|- (ps -> E.xE.yph)

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

Proof of Theorem cla4e2v

Step	Hyp	Ref	Expression
1		cla4e2v.1	. 2 |- A e. V
2		cla4e2v.2	. 2 |- B e. V
3		cla4e2v.3	. . 3 |- ((x = A /\ y = B) -> (ph <-> ps))
4	3	cla4e2gv 1398	. 2 |- ((A e. V /\ B e. V) -> (ps -> E.xE.yph))
5	1, 2, 4	mp2an 520	1 |- (ps -> E.xE.yph)

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  E.wex 678   = wceq 1091   e. wcel 1092  Vcvv 1348

This theorem is referenced by:  th3qlem2 3251  endisj 3341  axcnre 4087

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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