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Theorem copsex4g 4010
Description: An implicit substitution inference for 2 ordered pairs. (Contributed by NM, 5-Aug-1995.)
Hypothesis
Ref Expression
copsex4g.1  |-  ( ( ( x  =  A  /\  y  =  B )  /\  ( z  =  C  /\  w  =  D ) )  -> 
( ph  <->  ps ) )
Assertion
Ref Expression
copsex4g  |-  ( ( ( A  e.  R  /\  B  e.  S
)  /\  ( C  e.  R  /\  D  e.  S ) )  -> 
( E. x E. y E. z E. w
( ( <. A ,  B >.  =  <. x ,  y >.  /\  <. C ,  D >.  =  <. z ,  w >. )  /\  ph )  <->  ps )
)
Distinct variable groups:    x, y, z, w, A    x, B, y, z, w    x, C, y, z, w    x, D, y, z, w    ps, x, y, z, w    x, R, y, z, w    x, S, y, z, w
Allowed substitution hints:    ph( x, y, z, w)

Proof of Theorem copsex4g
StepHypRef Expression
1 eqcom 2084 . . . . . . 7  |-  ( <. A ,  B >.  = 
<. x ,  y >.  <->  <.
x ,  y >.  =  <. A ,  B >. )
2 vex 2605 . . . . . . . 8  |-  x  e. 
_V
3 vex 2605 . . . . . . . 8  |-  y  e. 
_V
42, 3opth 4000 . . . . . . 7  |-  ( <.
x ,  y >.  =  <. A ,  B >.  <-> 
( x  =  A  /\  y  =  B ) )
51, 4bitri 182 . . . . . 6  |-  ( <. A ,  B >.  = 
<. x ,  y >.  <->  ( x  =  A  /\  y  =  B )
)
6 eqcom 2084 . . . . . . 7  |-  ( <. C ,  D >.  = 
<. z ,  w >.  <->  <. z ,  w >.  =  <. C ,  D >. )
7 vex 2605 . . . . . . . 8  |-  z  e. 
_V
8 vex 2605 . . . . . . . 8  |-  w  e. 
_V
97, 8opth 4000 . . . . . . 7  |-  ( <.
z ,  w >.  = 
<. C ,  D >.  <->  (
z  =  C  /\  w  =  D )
)
106, 9bitri 182 . . . . . 6  |-  ( <. C ,  D >.  = 
<. z ,  w >.  <->  (
z  =  C  /\  w  =  D )
)
115, 10anbi12i 448 . . . . 5  |-  ( (
<. A ,  B >.  = 
<. x ,  y >.  /\  <. C ,  D >.  =  <. z ,  w >. )  <->  ( ( x  =  A  /\  y  =  B )  /\  (
z  =  C  /\  w  =  D )
) )
1211anbi1i 446 . . . 4  |-  ( ( ( <. A ,  B >.  =  <. x ,  y
>.  /\  <. C ,  D >.  =  <. z ,  w >. )  /\  ph )  <->  ( ( ( x  =  A  /\  y  =  B )  /\  (
z  =  C  /\  w  =  D )
)  /\  ph ) )
1312a1i 9 . . 3  |-  ( ( ( A  e.  R  /\  B  e.  S
)  /\  ( C  e.  R  /\  D  e.  S ) )  -> 
( ( ( <. A ,  B >.  = 
<. x ,  y >.  /\  <. C ,  D >.  =  <. z ,  w >. )  /\  ph )  <->  ( ( ( x  =  A  /\  y  =  B )  /\  (
z  =  C  /\  w  =  D )
)  /\  ph ) ) )
14134exbidv 1792 . 2  |-  ( ( ( A  e.  R  /\  B  e.  S
)  /\  ( C  e.  R  /\  D  e.  S ) )  -> 
( E. x E. y E. z E. w
( ( <. A ,  B >.  =  <. x ,  y >.  /\  <. C ,  D >.  =  <. z ,  w >. )  /\  ph )  <->  E. x E. y E. z E. w ( ( ( x  =  A  /\  y  =  B )  /\  ( z  =  C  /\  w  =  D ) )  /\  ph ) ) )
15 id 19 . . 3  |-  ( ( ( x  =  A  /\  y  =  B )  /\  ( z  =  C  /\  w  =  D ) )  -> 
( ( x  =  A  /\  y  =  B )  /\  (
z  =  C  /\  w  =  D )
) )
16 copsex4g.1 . . 3  |-  ( ( ( x  =  A  /\  y  =  B )  /\  ( z  =  C  /\  w  =  D ) )  -> 
( ph  <->  ps ) )
1715, 16cgsex4g 2637 . 2  |-  ( ( ( A  e.  R  /\  B  e.  S
)  /\  ( C  e.  R  /\  D  e.  S ) )  -> 
( E. x E. y E. z E. w
( ( ( x  =  A  /\  y  =  B )  /\  (
z  =  C  /\  w  =  D )
)  /\  ph )  <->  ps )
)
1814, 17bitrd 186 1  |-  ( ( ( A  e.  R  /\  B  e.  S
)  /\  ( C  e.  R  /\  D  e.  S ) )  -> 
( E. x E. y E. z E. w
( ( <. A ,  B >.  =  <. x ,  y >.  /\  <. C ,  D >.  =  <. z ,  w >. )  /\  ph )  <->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285   E.wex 1422    e. wcel 1434   <.cop 3409
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 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415
This theorem is referenced by:  opbrop  4445  ovi3  5668  dfplpq2  6606  dfmpq2  6607  enq0breq  6688
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