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Theorem prtlem13 29010
Description: Lemma for prter1 29021, prter2 29023, prter3 29024 and prtex 29022. (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 2973 . 2  |-  z  e. 
_V
2 vex 2973 . 2  |-  w  e. 
_V
3 elequ2 1761 . . . . 5  |-  ( u  =  v  ->  (
x  e.  u  <->  x  e.  v ) )
4 elequ2 1761 . . . . 5  |-  ( u  =  v  ->  (
y  e.  u  <->  y  e.  v ) )
53, 4anbi12d 710 . . . 4  |-  ( u  =  v  ->  (
( x  e.  u  /\  y  e.  u
)  <->  ( x  e.  v  /\  y  e.  v ) ) )
65cbvrexv 2946 . . 3  |-  ( E. u  e.  A  ( x  e.  u  /\  y  e.  u )  <->  E. v  e.  A  ( x  e.  v  /\  y  e.  v )
)
7 eleq1 2501 . . . . 5  |-  ( x  =  z  ->  (
x  e.  v  <->  z  e.  v ) )
8 eleq1 2501 . . . . 5  |-  ( y  =  w  ->  (
y  e.  v  <->  w  e.  v ) )
97, 8bi2anan9 868 . . . 4  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ( x  e.  v  /\  y  e.  v )  <->  ( z  e.  v  /\  w  e.  v ) ) )
109rexbidv 2734 . . 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 257 . 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 4604 1  |-  ( z  .~  w  <->  E. v  e.  A  ( z  e.  v  /\  w  e.  v ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369   E.wrex 2714   class class class wbr 4290   {copab 4347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pr 4529
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-br 4291  df-opab 4349
This theorem is referenced by:  prtlem16  29011  prtlem18  29019  prter1  29021  prter3  29024
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