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Theorem eqvinop 4226
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 4223 . . . . . . 7  |-  ( <.
x ,  y >.  =  <. B ,  C >.  <-> 
( x  =  B  /\  y  =  C ) )
43anbi2i 454 . . . . . 6  |-  ( ( A  =  <. x ,  y >.  /\  <. x ,  y >.  =  <. B ,  C >. )  <->  ( A  =  <. x ,  y >.  /\  (
x  =  B  /\  y  =  C )
) )
5 ancom 264 . . . . . 6  |-  ( ( A  =  <. x ,  y >.  /\  (
x  =  B  /\  y  =  C )
)  <->  ( ( x  =  B  /\  y  =  C )  /\  A  =  <. x ,  y
>. ) )
6 anass 399 . . . . . 6  |-  ( ( ( x  =  B  /\  y  =  C )  /\  A  = 
<. x ,  y >.
)  <->  ( x  =  B  /\  ( y  =  C  /\  A  =  <. x ,  y
>. ) ) )
74, 5, 63bitri 205 . . . . 5  |-  ( ( A  =  <. x ,  y >.  /\  <. x ,  y >.  =  <. B ,  C >. )  <->  ( x  =  B  /\  ( y  =  C  /\  A  =  <. x ,  y >. )
) )
87exbii 1598 . . . 4  |-  ( E. y ( A  = 
<. x ,  y >.  /\  <. x ,  y
>.  =  <. B ,  C >. )  <->  E. y
( x  =  B  /\  ( y  =  C  /\  A  = 
<. x ,  y >.
) ) )
9 19.42v 1899 . . . 4  |-  ( E. y ( x  =  B  /\  ( y  =  C  /\  A  =  <. x ,  y
>. ) )  <->  ( x  =  B  /\  E. y
( y  =  C  /\  A  =  <. x ,  y >. )
) )
10 opeq2 3764 . . . . . . 7  |-  ( y  =  C  ->  <. x ,  y >.  =  <. x ,  C >. )
1110eqeq2d 2182 . . . . . 6  |-  ( y  =  C  ->  ( A  =  <. x ,  y >.  <->  A  =  <. x ,  C >. )
)
122, 11ceqsexv 2769 . . . . 5  |-  ( E. y ( y  =  C  /\  A  = 
<. x ,  y >.
)  <->  A  =  <. x ,  C >. )
1312anbi2i 454 . . . 4  |-  ( ( x  =  B  /\  E. y ( y  =  C  /\  A  = 
<. x ,  y >.
) )  <->  ( x  =  B  /\  A  = 
<. x ,  C >. ) )
148, 9, 133bitri 205 . . 3  |-  ( E. y ( A  = 
<. x ,  y >.  /\  <. x ,  y
>.  =  <. B ,  C >. )  <->  ( x  =  B  /\  A  = 
<. x ,  C >. ) )
1514exbii 1598 . 2  |-  ( E. x E. y ( A  =  <. x ,  y >.  /\  <. x ,  y >.  =  <. B ,  C >. )  <->  E. x ( x  =  B  /\  A  = 
<. x ,  C >. ) )
16 opeq1 3763 . . . 4  |-  ( x  =  B  ->  <. x ,  C >.  =  <. B ,  C >. )
1716eqeq2d 2182 . . 3  |-  ( x  =  B  ->  ( A  =  <. x ,  C >.  <->  A  =  <. B ,  C >. )
)
181, 17ceqsexv 2769 . 2  |-  ( E. x ( x  =  B  /\  A  = 
<. x ,  C >. )  <-> 
A  =  <. B ,  C >. )
1915, 18bitr2i 184 1  |-  ( A  =  <. B ,  C >.  <->  E. x E. y ( A  =  <. x ,  y >.  /\  <. x ,  y >.  =  <. B ,  C >. )
)
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1348   E.wex 1485    e. wcel 2141   _Vcvv 2730   <.cop 3584
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-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590
This theorem is referenced by:  copsexg  4227  ralxpf  4755  rexxpf  4756  oprabid  5882
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