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Theorem eqvinop 4026
Description: A variable introduction law for ordered pairs. Analog of Lemma 15 of [Monk2] p. 109. (Contributed by NM, 28-May-1995.)
Hypotheses
Ref Expression
eqvinop.1  |-  B  e. 
_V
eqvinop.2  |-  C  e. 
_V
Assertion
Ref Expression
eqvinop  |-  ( A  =  <. B ,  C >.  <->  E. x E. y ( A  =  <. x ,  y >.  /\  <. x ,  y >.  =  <. B ,  C >. )
)
Distinct variable groups:    x, y, A   
x, B, y    x, C, y

Proof of Theorem eqvinop
StepHypRef Expression
1 eqvinop.1 . . . . . . . 8  |-  B  e. 
_V
2 eqvinop.2 . . . . . . . 8  |-  C  e. 
_V
31, 2opth2 4023 . . . . . . 7  |-  ( <.
x ,  y >.  =  <. B ,  C >.  <-> 
( x  =  B  /\  y  =  C ) )
43anbi2i 445 . . . . . 6  |-  ( ( A  =  <. x ,  y >.  /\  <. x ,  y >.  =  <. B ,  C >. )  <->  ( A  =  <. x ,  y >.  /\  (
x  =  B  /\  y  =  C )
) )
5 ancom 262 . . . . . 6  |-  ( ( A  =  <. x ,  y >.  /\  (
x  =  B  /\  y  =  C )
)  <->  ( ( x  =  B  /\  y  =  C )  /\  A  =  <. x ,  y
>. ) )
6 anass 393 . . . . . 6  |-  ( ( ( x  =  B  /\  y  =  C )  /\  A  = 
<. x ,  y >.
)  <->  ( x  =  B  /\  ( y  =  C  /\  A  =  <. x ,  y
>. ) ) )
74, 5, 63bitri 204 . . . . 5  |-  ( ( A  =  <. x ,  y >.  /\  <. x ,  y >.  =  <. B ,  C >. )  <->  ( x  =  B  /\  ( y  =  C  /\  A  =  <. x ,  y >. )
) )
87exbii 1537 . . . 4  |-  ( E. y ( A  = 
<. x ,  y >.  /\  <. x ,  y
>.  =  <. B ,  C >. )  <->  E. y
( x  =  B  /\  ( y  =  C  /\  A  = 
<. x ,  y >.
) ) )
9 19.42v 1829 . . . 4  |-  ( E. y ( x  =  B  /\  ( y  =  C  /\  A  =  <. x ,  y
>. ) )  <->  ( x  =  B  /\  E. y
( y  =  C  /\  A  =  <. x ,  y >. )
) )
10 opeq2 3591 . . . . . . 7  |-  ( y  =  C  ->  <. x ,  y >.  =  <. x ,  C >. )
1110eqeq2d 2094 . . . . . 6  |-  ( y  =  C  ->  ( A  =  <. x ,  y >.  <->  A  =  <. x ,  C >. )
)
122, 11ceqsexv 2647 . . . . 5  |-  ( E. y ( y  =  C  /\  A  = 
<. x ,  y >.
)  <->  A  =  <. x ,  C >. )
1312anbi2i 445 . . . 4  |-  ( ( x  =  B  /\  E. y ( y  =  C  /\  A  = 
<. x ,  y >.
) )  <->  ( x  =  B  /\  A  = 
<. x ,  C >. ) )
148, 9, 133bitri 204 . . 3  |-  ( E. y ( A  = 
<. x ,  y >.  /\  <. x ,  y
>.  =  <. B ,  C >. )  <->  ( x  =  B  /\  A  = 
<. x ,  C >. ) )
1514exbii 1537 . 2  |-  ( E. x E. y ( A  =  <. x ,  y >.  /\  <. x ,  y >.  =  <. B ,  C >. )  <->  E. x ( x  =  B  /\  A  = 
<. x ,  C >. ) )
16 opeq1 3590 . . . 4  |-  ( x  =  B  ->  <. x ,  C >.  =  <. B ,  C >. )
1716eqeq2d 2094 . . 3  |-  ( x  =  B  ->  ( A  =  <. x ,  C >.  <->  A  =  <. B ,  C >. )
)
181, 17ceqsexv 2647 . 2  |-  ( E. x ( x  =  B  /\  A  = 
<. x ,  C >. )  <-> 
A  =  <. B ,  C >. )
1915, 18bitr2i 183 1  |-  ( A  =  <. B ,  C >.  <->  E. x E. y ( A  =  <. x ,  y >.  /\  <. x ,  y >.  =  <. B ,  C >. )
)
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    = wceq 1285   E.wex 1422    e. wcel 1434   _Vcvv 2610   <.cop 3419
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425
This theorem is referenced by:  copsexg  4027  ralxpf  4530  rexxpf  4531  oprabid  5589
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