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Theorem spc2egv 3038
Description: Existential specialization with 2 quantifiers, using implicit substitution. (Contributed by NM, 3-Aug-1995.)
Hypothesis
Ref Expression
spc2egv.1  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
spc2egv  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ps  ->  E. x E. y ph ) )
Distinct variable groups:    x, y, A    x, B, y    ps, x, y
Allowed substitution hints:    ph( x, y)    V( x, y)    W( x, y)

Proof of Theorem spc2egv
StepHypRef Expression
1 elisset 2966 . . . 4  |-  ( A  e.  V  ->  E. x  x  =  A )
2 elisset 2966 . . . 4  |-  ( B  e.  W  ->  E. y 
y  =  B )
31, 2anim12i 550 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. x  x  =  A  /\  E. y  y  =  B
) )
4 eeanv 1937 . . 3  |-  ( E. x E. y ( x  =  A  /\  y  =  B )  <->  ( E. x  x  =  A  /\  E. y 
y  =  B ) )
53, 4sylibr 204 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  E. x E. y
( x  =  A  /\  y  =  B ) )
6 spc2egv.1 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
76biimprcd 217 . . 3  |-  ( ps 
->  ( ( x  =  A  /\  y  =  B )  ->  ph )
)
872eximdv 1634 . 2  |-  ( ps 
->  ( E. x E. y ( x  =  A  /\  y  =  B )  ->  E. x E. y ph ) )
95, 8syl5com 28 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ps  ->  E. x E. y ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725
This theorem is referenced by:  spc2gv  3039  spc2ev  3044  th3q  7013  0pthonv  21581  1pthon2v  21593  2pthon3v  21604  usg2wlk  28319  usg2wlkon  28320
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-v 2958
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