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Theorem spc2ev 3141
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 3135 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( ps  ->  E. x E. y ph ) )
51, 2, 4mp2an 677 1  |-  ( ps 
->  E. x E. y ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1443   E.wex 1662    e. wcel 1886   _Vcvv 3044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-10 1914  ax-11 1919  ax-12 1932  ax-ext 2430
This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-clab 2437  df-cleq 2443  df-clel 2446  df-v 3046
This theorem is referenced by:  relop  4984  endisj  7656  dcomex  8874  axcnre  9585  constr3cyclpe  25384  3v3e3cycl2  25385  qqhval2  28779  itg2addnclem3  31988  funop1  39014  wlkc  39651
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