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Theorem eqvinop 4686
Description: A variable introduction law for ordered pairs. Analogue 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 4680 . . . . . . 7  |-  ( <.
x ,  y >.  =  <. B ,  C >.  <-> 
( x  =  B  /\  y  =  C ) )
43anbi2i 708 . . . . . 6  |-  ( ( A  =  <. x ,  y >.  /\  <. x ,  y >.  =  <. B ,  C >. )  <->  ( A  =  <. x ,  y >.  /\  (
x  =  B  /\  y  =  C )
) )
5 ancom 457 . . . . . 6  |-  ( ( A  =  <. x ,  y >.  /\  (
x  =  B  /\  y  =  C )
)  <->  ( ( x  =  B  /\  y  =  C )  /\  A  =  <. x ,  y
>. ) )
6 anass 661 . . . . . 6  |-  ( ( ( x  =  B  /\  y  =  C )  /\  A  = 
<. x ,  y >.
)  <->  ( x  =  B  /\  ( y  =  C  /\  A  =  <. x ,  y
>. ) ) )
74, 5, 63bitri 279 . . . . 5  |-  ( ( A  =  <. x ,  y >.  /\  <. x ,  y >.  =  <. B ,  C >. )  <->  ( x  =  B  /\  ( y  =  C  /\  A  =  <. x ,  y >. )
) )
87exbii 1726 . . . 4  |-  ( E. y ( A  = 
<. x ,  y >.  /\  <. x ,  y
>.  =  <. B ,  C >. )  <->  E. y
( x  =  B  /\  ( y  =  C  /\  A  = 
<. x ,  y >.
) ) )
9 19.42v 1842 . . . 4  |-  ( E. y ( x  =  B  /\  ( y  =  C  /\  A  =  <. x ,  y
>. ) )  <->  ( x  =  B  /\  E. y
( y  =  C  /\  A  =  <. x ,  y >. )
) )
10 opeq2 4159 . . . . . . 7  |-  ( y  =  C  ->  <. x ,  y >.  =  <. x ,  C >. )
1110eqeq2d 2481 . . . . . 6  |-  ( y  =  C  ->  ( A  =  <. x ,  y >.  <->  A  =  <. x ,  C >. )
)
122, 11ceqsexv 3070 . . . . 5  |-  ( E. y ( y  =  C  /\  A  = 
<. x ,  y >.
)  <->  A  =  <. x ,  C >. )
1312anbi2i 708 . . . 4  |-  ( ( x  =  B  /\  E. y ( y  =  C  /\  A  = 
<. x ,  y >.
) )  <->  ( x  =  B  /\  A  = 
<. x ,  C >. ) )
148, 9, 133bitri 279 . . 3  |-  ( E. y ( A  = 
<. x ,  y >.  /\  <. x ,  y
>.  =  <. B ,  C >. )  <->  ( x  =  B  /\  A  = 
<. x ,  C >. ) )
1514exbii 1726 . 2  |-  ( E. x E. y ( A  =  <. x ,  y >.  /\  <. x ,  y >.  =  <. B ,  C >. )  <->  E. x ( x  =  B  /\  A  = 
<. x ,  C >. ) )
16 opeq1 4158 . . . 4  |-  ( x  =  B  ->  <. x ,  C >.  =  <. B ,  C >. )
1716eqeq2d 2481 . . 3  |-  ( x  =  B  ->  ( A  =  <. x ,  C >.  <->  A  =  <. B ,  C >. )
)
181, 17ceqsexv 3070 . 2  |-  ( E. x ( x  =  B  /\  A  = 
<. x ,  C >. )  <-> 
A  =  <. B ,  C >. )
1915, 18bitr2i 258 1  |-  ( A  =  <. B ,  C >.  <->  E. x E. y ( A  =  <. x ,  y >.  /\  <. x ,  y >.  =  <. B ,  C >. )
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 376    = wceq 1452   E.wex 1671    e. wcel 1904   _Vcvv 3031   <.cop 3965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966
This theorem is referenced by:  copsexg  4687  ralxpf  4986  oprabid  6335
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