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Theorem prtlem13 26708
Description: Lemma for prter1 26719, prter2 26721, prter3 26722 and prtex 26720. (Contributed by Rodolfo Medina, 13-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypothesis
Ref Expression
prtlem13.1  |-  .~  =  { <. x ,  y
>.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u ) }
Assertion
Ref Expression
prtlem13  |-  ( z  .~  w  <->  E. v  e.  A  ( z  e.  v  /\  w  e.  v ) )
Distinct variable groups:    v, u, x, y, A    w, v, x, y    z, v, x, y
Allowed substitution hints:    A( z, w)    .~ ( x, y, z, w, v, u)

Proof of Theorem prtlem13
StepHypRef Expression
1 vex 2951 . 2  |-  z  e. 
_V
2 vex 2951 . 2  |-  w  e. 
_V
3 elequ2 1730 . . . . 5  |-  ( u  =  v  ->  (
x  e.  u  <->  x  e.  v ) )
4 elequ2 1730 . . . . 5  |-  ( u  =  v  ->  (
y  e.  u  <->  y  e.  v ) )
53, 4anbi12d 692 . . . 4  |-  ( u  =  v  ->  (
( x  e.  u  /\  y  e.  u
)  <->  ( x  e.  v  /\  y  e.  v ) ) )
65cbvrexv 2925 . . 3  |-  ( E. u  e.  A  ( x  e.  u  /\  y  e.  u )  <->  E. v  e.  A  ( x  e.  v  /\  y  e.  v )
)
7 eleq1 2495 . . . . 5  |-  ( x  =  z  ->  (
x  e.  v  <->  z  e.  v ) )
8 eleq1 2495 . . . . 5  |-  ( y  =  w  ->  (
y  e.  v  <->  w  e.  v ) )
97, 8bi2anan9 844 . . . 4  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ( x  e.  v  /\  y  e.  v )  <->  ( z  e.  v  /\  w  e.  v ) ) )
109rexbidv 2718 . . 3  |-  ( ( x  =  z  /\  y  =  w )  ->  ( E. v  e.  A  ( x  e.  v  /\  y  e.  v )  <->  E. v  e.  A  ( z  e.  v  /\  w  e.  v ) ) )
116, 10syl5bb 249 . 2  |-  ( ( x  =  z  /\  y  =  w )  ->  ( E. u  e.  A  ( x  e.  u  /\  y  e.  u )  <->  E. v  e.  A  ( z  e.  v  /\  w  e.  v ) ) )
12 prtlem13.1 . 2  |-  .~  =  { <. x ,  y
>.  |  E. u  e.  A  ( x  e.  u  /\  y  e.  u ) }
131, 2, 11, 12braba 4464 1  |-  ( z  .~  w  <->  E. v  e.  A  ( z  e.  v  /\  w  e.  v ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652   E.wrex 2698   class class class wbr 4204   {copab 4257
This theorem is referenced by:  prtlem16  26709  prtlem18  26717  prter1  26719  prter3  26722
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-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  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-br 4205  df-opab 4259
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