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Theorem eqvinop 4250
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 4247 . . . . . . 7  |-  ( <.
x ,  y >.  =  <. B ,  C >.  <-> 
( x  =  B  /\  y  =  C ) )
43anbi2i 675 . . . . . 6  |-  ( ( A  =  <. x ,  y >.  /\  <. x ,  y >.  =  <. B ,  C >. )  <->  ( A  =  <. x ,  y >.  /\  (
x  =  B  /\  y  =  C )
) )
5 ancom 437 . . . . . 6  |-  ( ( A  =  <. x ,  y >.  /\  (
x  =  B  /\  y  =  C )
)  <->  ( ( x  =  B  /\  y  =  C )  /\  A  =  <. x ,  y
>. ) )
6 anass 630 . . . . . 6  |-  ( ( ( x  =  B  /\  y  =  C )  /\  A  = 
<. x ,  y >.
)  <->  ( x  =  B  /\  ( y  =  C  /\  A  =  <. x ,  y
>. ) ) )
74, 5, 63bitri 262 . . . . 5  |-  ( ( A  =  <. x ,  y >.  /\  <. x ,  y >.  =  <. B ,  C >. )  <->  ( x  =  B  /\  ( y  =  C  /\  A  =  <. x ,  y >. )
) )
87exbii 1569 . . . 4  |-  ( E. y ( A  = 
<. x ,  y >.  /\  <. x ,  y
>.  =  <. B ,  C >. )  <->  E. y
( x  =  B  /\  ( y  =  C  /\  A  = 
<. x ,  y >.
) ) )
9 19.42v 1848 . . . 4  |-  ( E. y ( x  =  B  /\  ( y  =  C  /\  A  =  <. x ,  y
>. ) )  <->  ( x  =  B  /\  E. y
( y  =  C  /\  A  =  <. x ,  y >. )
) )
10 opeq2 3798 . . . . . . 7  |-  ( y  =  C  ->  <. x ,  y >.  =  <. x ,  C >. )
1110eqeq2d 2295 . . . . . 6  |-  ( y  =  C  ->  ( A  =  <. x ,  y >.  <->  A  =  <. x ,  C >. )
)
122, 11ceqsexv 2824 . . . . 5  |-  ( E. y ( y  =  C  /\  A  = 
<. x ,  y >.
)  <->  A  =  <. x ,  C >. )
1312anbi2i 675 . . . 4  |-  ( ( x  =  B  /\  E. y ( y  =  C  /\  A  = 
<. x ,  y >.
) )  <->  ( x  =  B  /\  A  = 
<. x ,  C >. ) )
148, 9, 133bitri 262 . . 3  |-  ( E. y ( A  = 
<. x ,  y >.  /\  <. x ,  y
>.  =  <. B ,  C >. )  <->  ( x  =  B  /\  A  = 
<. x ,  C >. ) )
1514exbii 1569 . 2  |-  ( E. x E. y ( A  =  <. x ,  y >.  /\  <. x ,  y >.  =  <. B ,  C >. )  <->  E. x ( x  =  B  /\  A  = 
<. x ,  C >. ) )
16 opeq1 3797 . . . 4  |-  ( x  =  B  ->  <. x ,  C >.  =  <. B ,  C >. )
1716eqeq2d 2295 . . 3  |-  ( x  =  B  ->  ( A  =  <. x ,  C >.  <->  A  =  <. B ,  C >. )
)
181, 17ceqsexv 2824 . 2  |-  ( E. x ( x  =  B  /\  A  = 
<. x ,  C >. )  <-> 
A  =  <. B ,  C >. )
1915, 18bitr2i 241 1  |-  ( A  =  <. B ,  C >.  <->  E. x E. y ( A  =  <. x ,  y >.  /\  <. x ,  y >.  =  <. B ,  C >. )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1685   _Vcvv 2789   <.cop 3644
This theorem is referenced by:  copsexg  4251  ralxpf  4829  oprabid  5844
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650
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