MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  copsex2g Unicode version

Theorem copsex2g 4191
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 2750 . 2  |-  ( A  e.  V  ->  E. x  x  =  A )
2 elisset 2750 . 2  |-  ( B  e.  W  ->  E. y 
y  =  B )
3 eeanv 2058 . . 3  |-  ( E. x E. y ( x  =  A  /\  y  =  B )  <->  ( E. x  x  =  A  /\  E. y 
y  =  B ) )
4 nfe1 1566 . . . . 5  |-  F/ x E. x E. y (
<. A ,  B >.  = 
<. x ,  y >.  /\  ph )
5 nfv 1629 . . . . 5  |-  F/ x ps
64, 5nfbi 1738 . . . 4  |-  F/ x
( E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) 
<->  ps )
7 nfe1 1566 . . . . . . 7  |-  F/ y E. y ( <. A ,  B >.  = 
<. x ,  y >.  /\  ph )
87nfex 1733 . . . . . 6  |-  F/ y E. x E. y
( <. A ,  B >.  =  <. x ,  y
>.  /\  ph )
9 nfv 1629 . . . . . 6  |-  F/ y ps
108, 9nfbi 1738 . . . . 5  |-  F/ y ( E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) 
<->  ps )
11 opeq12 3739 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  -> 
<. x ,  y >.  =  <. A ,  B >. )
12 copsexg 4189 . . . . . . . 8  |-  ( <. A ,  B >.  = 
<. x ,  y >.  ->  ( ph  <->  E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) ) )
1312eqcoms 2259 . . . . . . 7  |-  ( <.
x ,  y >.  =  <. A ,  B >.  ->  ( ph  <->  E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) ) )
1411, 13syl 17 . . . . . 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 248 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) 
<->  ps ) )
1710, 16exlimi 1781 . . . 4  |-  ( E. y ( x  =  A  /\  y  =  B )  ->  ( E. x E. y (
<. A ,  B >.  = 
<. x ,  y >.  /\  ph )  <->  ps )
)
186, 17exlimi 1781 . . 3  |-  ( E. x E. y ( x  =  A  /\  y  =  B )  ->  ( E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) 
<->  ps ) )
193, 18sylbir 206 . 2  |-  ( ( E. x  x  =  A  /\  E. y 
y  =  B )  ->  ( E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) 
<->  ps ) )
201, 2, 19syl2an 465 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 6    <-> wb 178    /\ wa 360   E.wex 1537    = wceq 1619    e. wcel 1621   <.cop 3584
This theorem is referenced by:  opelopabga  4215  ov6g  5884  ltresr  8695
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-rab 2523  df-v 2742  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590
  Copyright terms: Public domain W3C validator