Users' Mathboxes Mathbox for Rodolfo Medina < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  prtlem13 Structured version   Unicode version

Theorem prtlem13 30771
Description: Lemma for prter1 30782, prter2 30784, prter3 30785 and prtex 30783. (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 3112 . 2  |-  z  e. 
_V
2 vex 3112 . 2  |-  w  e. 
_V
3 elequ2 1824 . . . . 5  |-  ( u  =  v  ->  (
x  e.  u  <->  x  e.  v ) )
4 elequ2 1824 . . . . 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 3085 . . 3  |-  ( E. u  e.  A  ( x  e.  u  /\  y  e.  u )  <->  E. v  e.  A  ( x  e.  v  /\  y  e.  v )
)
7 eleq1 2529 . . . . 5  |-  ( x  =  z  ->  (
x  e.  v  <->  z  e.  v ) )
8 eleq1 2529 . . . . 5  |-  ( y  =  w  ->  (
y  e.  v  <->  w  e.  v ) )
97, 8bi2anan9 873 . . . 4  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ( x  e.  v  /\  y  e.  v )  <->  ( z  e.  v  /\  w  e.  v ) ) )
109rexbidv 2968 . . 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 4773 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 1395   E.wrex 2808   class class class wbr 4456   {copab 4514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516
This theorem is referenced by:  prtlem16  30772  prtlem18  30780  prter1  30782  prter3  30785
  Copyright terms: Public domain W3C validator