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Theorem spc2ev 3163
Description: Existential specialization, using implicit substitution. (Contributed by NM, 3-Aug-1995.)
Hypotheses
Ref Expression
spc2ev.1  |-  A  e. 
_V
spc2ev.2  |-  B  e. 
_V
spc2ev.3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
spc2ev  |-  ( ps 
->  E. x E. y ph )
Distinct variable groups:    x, y, A    x, B, y    ps, x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem spc2ev
StepHypRef Expression
1 spc2ev.1 . 2  |-  A  e. 
_V
2 spc2ev.2 . 2  |-  B  e. 
_V
3 spc2ev.3 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
43spc2egv 3157 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( ps  ->  E. x E. y ph ) )
51, 2, 4mp2an 672 1  |-  ( ps 
->  E. x E. y ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370   E.wex 1587    e. wcel 1758   _Vcvv 3070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-v 3072
This theorem is referenced by:  relop  5090  th3qlem2  7309  endisj  7500  dcomex  8719  axcnre  9434  constr3cyclpe  23686  3v3e3cycl2  23687  qqhval2  26547  itg2addnclem3  28585
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