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Theorem copsex2g 4270
Description: Implicit substitution inference for ordered pairs. (Contributed by NM, 28-May-1995.)
Hypothesis
Ref Expression
copsex2g.1  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
copsex2g  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) 
<->  ps ) )
Distinct variable groups:    x, y, ps    x, A, y    x, B, y
Allowed substitution hints:    ph( x, y)    V( x, y)    W( x, y)

Proof of Theorem copsex2g
StepHypRef Expression
1 elisset 2811 . 2  |-  ( A  e.  V  ->  E. x  x  =  A )
2 elisset 2811 . 2  |-  ( B  e.  W  ->  E. y 
y  =  B )
3 eeanv 1866 . . 3  |-  ( E. x E. y ( x  =  A  /\  y  =  B )  <->  ( E. x  x  =  A  /\  E. y 
y  =  B ) )
4 nfe1 1718 . . . . 5  |-  F/ x E. x E. y (
<. A ,  B >.  = 
<. x ,  y >.  /\  ph )
5 nfv 1609 . . . . 5  |-  F/ x ps
64, 5nfbi 1784 . . . 4  |-  F/ x
( E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) 
<->  ps )
7 nfe1 1718 . . . . . . 7  |-  F/ y E. y ( <. A ,  B >.  = 
<. x ,  y >.  /\  ph )
87nfex 1779 . . . . . 6  |-  F/ y E. x E. y
( <. A ,  B >.  =  <. x ,  y
>.  /\  ph )
9 nfv 1609 . . . . . 6  |-  F/ y ps
108, 9nfbi 1784 . . . . 5  |-  F/ y ( E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) 
<->  ps )
11 opeq12 3814 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  -> 
<. x ,  y >.  =  <. A ,  B >. )
12 copsexg 4268 . . . . . . . 8  |-  ( <. A ,  B >.  = 
<. x ,  y >.  ->  ( ph  <->  E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) ) )
1312eqcoms 2299 . . . . . . 7  |-  ( <.
x ,  y >.  =  <. A ,  B >.  ->  ( ph  <->  E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) ) )
1411, 13syl 15 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) ) )
15 copsex2g.1 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
1614, 15bitr3d 246 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) 
<->  ps ) )
1710, 16exlimi 1813 . . . 4  |-  ( E. y ( x  =  A  /\  y  =  B )  ->  ( E. x E. y (
<. A ,  B >.  = 
<. x ,  y >.  /\  ph )  <->  ps )
)
186, 17exlimi 1813 . . 3  |-  ( E. x E. y ( x  =  A  /\  y  =  B )  ->  ( E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) 
<->  ps ) )
193, 18sylbir 204 . 2  |-  ( ( E. x  x  =  A  /\  E. y 
y  =  B )  ->  ( E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) 
<->  ps ) )
201, 2, 19syl2an 463 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) 
<->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   <.cop 3656
This theorem is referenced by:  opelopabga  4294  ov6g  6001  ltresr  8778
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662
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