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Theorem spc2ev 2889
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 2883 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( ps  ->  E. x E. y ph ) )
51, 2, 4mp2an 653 1  |-  ( ps 
->  E. x E. y ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801
This theorem is referenced by:  relop  4850  th3qlem2  6781  endisj  6965  dcomex  8089  axcnre  8802  constr3cyclpe  28409  3v3e3cycl2  28410
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-v 2803
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