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Theorem eqvinop 4401
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 4398 . . . . . . 7  |-  ( <.
x ,  y >.  =  <. B ,  C >.  <-> 
( x  =  B  /\  y  =  C ) )
43anbi2i 676 . . . . . 6  |-  ( ( A  =  <. x ,  y >.  /\  <. x ,  y >.  =  <. B ,  C >. )  <->  ( A  =  <. x ,  y >.  /\  (
x  =  B  /\  y  =  C )
) )
5 ancom 438 . . . . . 6  |-  ( ( A  =  <. x ,  y >.  /\  (
x  =  B  /\  y  =  C )
)  <->  ( ( x  =  B  /\  y  =  C )  /\  A  =  <. x ,  y
>. ) )
6 anass 631 . . . . . 6  |-  ( ( ( x  =  B  /\  y  =  C )  /\  A  = 
<. x ,  y >.
)  <->  ( x  =  B  /\  ( y  =  C  /\  A  =  <. x ,  y
>. ) ) )
74, 5, 63bitri 263 . . . . 5  |-  ( ( A  =  <. x ,  y >.  /\  <. x ,  y >.  =  <. B ,  C >. )  <->  ( x  =  B  /\  ( y  =  C  /\  A  =  <. x ,  y >. )
) )
87exbii 1589 . . . 4  |-  ( E. y ( A  = 
<. x ,  y >.  /\  <. x ,  y
>.  =  <. B ,  C >. )  <->  E. y
( x  =  B  /\  ( y  =  C  /\  A  = 
<. x ,  y >.
) ) )
9 19.42v 1924 . . . 4  |-  ( E. y ( x  =  B  /\  ( y  =  C  /\  A  =  <. x ,  y
>. ) )  <->  ( x  =  B  /\  E. y
( y  =  C  /\  A  =  <. x ,  y >. )
) )
10 opeq2 3945 . . . . . . 7  |-  ( y  =  C  ->  <. x ,  y >.  =  <. x ,  C >. )
1110eqeq2d 2415 . . . . . 6  |-  ( y  =  C  ->  ( A  =  <. x ,  y >.  <->  A  =  <. x ,  C >. )
)
122, 11ceqsexv 2951 . . . . 5  |-  ( E. y ( y  =  C  /\  A  = 
<. x ,  y >.
)  <->  A  =  <. x ,  C >. )
1312anbi2i 676 . . . 4  |-  ( ( x  =  B  /\  E. y ( y  =  C  /\  A  = 
<. x ,  y >.
) )  <->  ( x  =  B  /\  A  = 
<. x ,  C >. ) )
148, 9, 133bitri 263 . . 3  |-  ( E. y ( A  = 
<. x ,  y >.  /\  <. x ,  y
>.  =  <. B ,  C >. )  <->  ( x  =  B  /\  A  = 
<. x ,  C >. ) )
1514exbii 1589 . 2  |-  ( E. x E. y ( A  =  <. x ,  y >.  /\  <. x ,  y >.  =  <. B ,  C >. )  <->  E. x ( x  =  B  /\  A  = 
<. x ,  C >. ) )
16 opeq1 3944 . . . 4  |-  ( x  =  B  ->  <. x ,  C >.  =  <. B ,  C >. )
1716eqeq2d 2415 . . 3  |-  ( x  =  B  ->  ( A  =  <. x ,  C >.  <->  A  =  <. B ,  C >. )
)
181, 17ceqsexv 2951 . 2  |-  ( E. x ( x  =  B  /\  A  = 
<. x ,  C >. )  <-> 
A  =  <. B ,  C >. )
1915, 18bitr2i 242 1  |-  ( A  =  <. B ,  C >.  <->  E. x E. y ( A  =  <. x ,  y >.  /\  <. x ,  y >.  =  <. B ,  C >. )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721   _Vcvv 2916   <.cop 3777
This theorem is referenced by:  copsexg  4402  ralxpf  4978  oprabid  6064
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783
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