Theorem cla42ev 1873

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
cla42ev	|- (ps -> E.xE.yph)

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

Proof of Theorem cla42ev

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	cla42egv 1867	. 2 |- ((A e. V /\ B e. V) -> (ps -> E.xE.yph))
5	1, 2, 4	mp2an 699	1 |- (ps -> E.xE.yph)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  E.wex 982  Vcvv 1814

This theorem is referenced by:  relop 3281  th3qlem2 4321  endisj 4443  axcnre 5298

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815

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