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Theorem opelopab4 28575
Description: Ordered pair membership in a class abstraction of pairs. Compare to elopab 4454. (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 4454 . 2  |-  ( <.
u ,  v >.  e.  { <. x ,  y
>.  |  ph }  <->  E. x E. y ( <. u ,  v >.  =  <. x ,  y >.  /\  ph ) )
2 vex 2951 . . . . . 6  |-  x  e. 
_V
3 vex 2951 . . . . . 6  |-  y  e. 
_V
42, 3opth 4427 . . . . 5  |-  ( <.
x ,  y >.  =  <. u ,  v
>. 
<->  ( x  =  u  /\  y  =  v ) )
5 eqcom 2437 . . . . 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 1593 . 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 1550    = wceq 1652    e. wcel 1725   <.cop 3809   {copab 4257
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-opab 4259
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