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Theorem prtlem13 26401
Description: Lemma for prter1 26412, prter2 26414, prter3 26415 and prtex 26413. (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 2895 . 2  |-  z  e. 
_V
2 vex 2895 . 2  |-  w  e. 
_V
3 elequ2 1722 . . . . 5  |-  ( u  =  v  ->  (
x  e.  u  <->  x  e.  v ) )
4 elequ2 1722 . . . . 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 2869 . . 3  |-  ( E. u  e.  A  ( x  e.  u  /\  y  e.  u )  <->  E. v  e.  A  ( x  e.  v  /\  y  e.  v )
)
7 eleq1 2440 . . . . 5  |-  ( x  =  z  ->  (
x  e.  v  <->  z  e.  v ) )
8 eleq1 2440 . . . . 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 2663 . . 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 4406 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 1649   E.wrex 2643   class class class wbr 4146   {copab 4199
This theorem is referenced by:  prtlem16  26402  prtlem18  26410  prter1  26412  prter3  26415
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-br 4147  df-opab 4201
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