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Theorem opelopab4 36961
Description: Ordered pair membership in a class abstraction of pairs. Compare to elopab 4722. (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 4722 . 2  |-  ( <.
u ,  v >.  e.  { <. x ,  y
>.  |  ph }  <->  E. x E. y ( <. u ,  v >.  =  <. x ,  y >.  /\  ph ) )
2 vex 3059 . . . . . 6  |-  x  e. 
_V
3 vex 3059 . . . . . 6  |-  y  e. 
_V
42, 3opth 4689 . . . . 5  |-  ( <.
x ,  y >.  =  <. u ,  v
>. 
<->  ( x  =  u  /\  y  =  v ) )
5 eqcom 2468 . . . . 5  |-  ( <.
x ,  y >.  =  <. u ,  v
>. 
<-> 
<. u ,  v >.  =  <. x ,  y
>. )
64, 5bitr3i 259 . . . 4  |-  ( ( x  =  u  /\  y  =  v )  <->  <.
u ,  v >.  =  <. x ,  y
>. )
76anbi1i 706 . . 3  |-  ( ( ( x  =  u  /\  y  =  v )  /\  ph )  <->  (
<. u ,  v >.  =  <. x ,  y
>.  /\  ph ) )
872exbii 1729 . 2  |-  ( E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ph )  <->  E. x E. y (
<. u ,  v >.  =  <. x ,  y
>.  /\  ph ) )
91, 8bitr4i 260 1  |-  ( <.
u ,  v >.  e.  { <. x ,  y
>.  |  ph }  <->  E. x E. y ( ( x  =  u  /\  y  =  v )  /\  ph ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 375    = wceq 1454   E.wex 1673    e. wcel 1897   <.cop 3985   {copab 4473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-rab 2757  df-v 3058  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-opab 4475
This theorem is referenced by: (None)
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