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Theorem opelopab4 28317
Description: Ordered pair membership in a class abstraction of pairs. Compare to elopab 4272. (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 4272 . 2  |-  ( <.
u ,  v >.  e.  { <. x ,  y
>.  |  ph }  <->  E. x E. y ( <. u ,  v >.  =  <. x ,  y >.  /\  ph ) )
2 vex 2791 . . . . . 6  |-  x  e. 
_V
3 vex 2791 . . . . . 6  |-  y  e. 
_V
42, 3opth 4245 . . . . 5  |-  ( <.
x ,  y >.  =  <. u ,  v
>. 
<->  ( x  =  u  /\  y  =  v ) )
5 eqcom 2285 . . . . 5  |-  ( <.
x ,  y >.  =  <. u ,  v
>. 
<-> 
<. u ,  v >.  =  <. x ,  y
>. )
64, 5bitr3i 242 . . . 4  |-  ( ( x  =  u  /\  y  =  v )  <->  <.
u ,  v >.  =  <. x ,  y
>. )
76anbi1i 676 . . 3  |-  ( ( ( x  =  u  /\  y  =  v )  /\  ph )  <->  (
<. u ,  v >.  =  <. x ,  y
>.  /\  ph ) )
872exbii 1570 . 2  |-  ( E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ph )  <->  E. x E. y (
<. u ,  v >.  =  <. x ,  y
>.  /\  ph ) )
91, 8bitr4i 243 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 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   <.cop 3643   {copab 4076
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 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-opab 4078
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