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Theorem prtlem13 26839
Description: Lemma for prter1 26850, prter2 26852, prter3 26853 and prtex 26851. (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 2804 . 2  |-  z  e. 
_V
2 vex 2804 . 2  |-  w  e. 
_V
3 elequ2 1701 . . . . 5  |-  ( u  =  v  ->  (
x  e.  u  <->  x  e.  v ) )
4 elequ2 1701 . . . . 5  |-  ( u  =  v  ->  (
y  e.  u  <->  y  e.  v ) )
53, 4anbi12d 691 . . . 4  |-  ( u  =  v  ->  (
( x  e.  u  /\  y  e.  u
)  <->  ( x  e.  v  /\  y  e.  v ) ) )
65cbvrexv 2778 . . 3  |-  ( E. u  e.  A  ( x  e.  u  /\  y  e.  u )  <->  E. v  e.  A  ( x  e.  v  /\  y  e.  v )
)
7 eleq1 2356 . . . . 5  |-  ( x  =  z  ->  (
x  e.  v  <->  z  e.  v ) )
8 eleq1 2356 . . . . 5  |-  ( y  =  w  ->  (
y  e.  v  <->  w  e.  v ) )
97, 8bi2anan9 843 . . . 4  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ( x  e.  v  /\  y  e.  v )  <->  ( z  e.  v  /\  w  e.  v ) ) )
109rexbidv 2577 . . 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 248 . 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 4298 1  |-  ( z  .~  w  <->  E. v  e.  A  ( z  e.  v  /\  w  e.  v ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557   class class class wbr 4039   {copab 4092
This theorem is referenced by:  prtlem16  26840  prtlem18  26848  prter1  26850  prter3  26853
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094
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