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Theorem spc2egv 2747
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 2672 . . . 4  |-  ( A  e.  V  ->  E. x  x  =  A )
2 elisset 2672 . . . 4  |-  ( B  e.  W  ->  E. y 
y  =  B )
31, 2anim12i 334 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. x  x  =  A  /\  E. y  y  =  B
) )
4 eeanv 1882 . . 3  |-  ( E. x E. y ( x  =  A  /\  y  =  B )  <->  ( E. x  x  =  A  /\  E. y 
y  =  B ) )
53, 4sylibr 133 . 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 159 . . 3  |-  ( ps 
->  ( ( x  =  A  /\  y  =  B )  ->  ph )
)
872eximdv 1836 . 2  |-  ( ps 
->  ( E. x E. y ( x  =  A  /\  y  =  B )  ->  E. x E. y ph ) )
95, 8syl5com 29 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ps  ->  E. x E. y ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314   E.wex 1451    e. wcel 1463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-v 2660
This theorem is referenced by:  spc2ev  2753  th3q  6500  addnnnq0  7221  mulnnnq0  7222  addsrpr  7517  mulsrpr  7518
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