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Theorem opelopab4 27983
Description: Ordered pair membership in a class abstraction of pairs. Compare to elopab 4405. (Contributed by Alan Sare, 8-Feb-2014.) (Revised by Mario Carneiro, 6-May-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
opelopab4  |-  ( <.
u ,  v >.  e.  { <. x ,  y
>.  |  ph }  <->  E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ph ) )
Distinct variable groups:    x, u    y, u    x, v    y,
v
Allowed substitution hints:    ph( x, y, v, u)

Proof of Theorem opelopab4
StepHypRef Expression
1 elopab 4405 . 2  |-  ( <.
u ,  v >.  e.  { <. x ,  y
>.  |  ph }  <->  E. x E. y ( <. u ,  v >.  =  <. x ,  y >.  /\  ph ) )
2 vex 2904 . . . . . 6  |-  x  e. 
_V
3 vex 2904 . . . . . 6  |-  y  e. 
_V
42, 3opth 4378 . . . . 5  |-  ( <.
x ,  y >.  =  <. u ,  v
>. 
<->  ( x  =  u  /\  y  =  v ) )
5 eqcom 2391 . . . . 5  |-  ( <.
x ,  y >.  =  <. u ,  v
>. 
<-> 
<. u ,  v >.  =  <. x ,  y
>. )
64, 5bitr3i 243 . . . 4  |-  ( ( x  =  u  /\  y  =  v )  <->  <.
u ,  v >.  =  <. x ,  y
>. )
76anbi1i 677 . . 3  |-  ( ( ( x  =  u  /\  y  =  v )  /\  ph )  <->  (
<. u ,  v >.  =  <. x ,  y
>.  /\  ph ) )
872exbii 1590 . 2  |-  ( E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ph )  <->  E. x E. y (
<. u ,  v >.  =  <. x ,  y
>.  /\  ph ) )
91, 8bitr4i 244 1  |-  ( <.
u ,  v >.  e.  { <. x ,  y
>.  |  ph }  <->  E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ph ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717   <.cop 3762   {copab 4208
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-opab 4210
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