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Theorem ltdfpr 6489
Description: More convenient form of df-iltp 6453. (Contributed by Jim Kingdon, 15-Dec-2019.)
Assertion
Ref Expression
ltdfpr  P.  P.  <P  q  Q.  q  2nd `  q  1st `
Distinct variable groups:   , q   , q

Proof of Theorem ltdfpr
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-br 3756 . . 3 
<P  <. ,  >.  <P
2 df-iltp 6453 . . . 4  <P  { <. , 
>.  |  P.  P.  q  Q. 
q  2nd `  q  1st `  }
32eleq2i 2101 . . 3  <. ,  >.  <P  <. ,  >. 
{ <. , 
>.  |  P.  P.  q  Q. 
q  2nd `  q  1st `  }
41, 3bitri 173 . 2 
<P  <. ,  >.  { <. ,  >.  |  P.  P.  q  Q.  q  2nd `  q  1st `  }
5 simpl 102 . . . . . . 7
65fveq2d 5125 . . . . . 6  2nd `  2nd `
76eleq2d 2104 . . . . 5  q  2nd `  q  2nd `
8 simpr 103 . . . . . . 7
98fveq2d 5125 . . . . . 6  1st `  1st `
109eleq2d 2104 . . . . 5  q  1st `  q  1st `
117, 10anbi12d 442 . . . 4  q  2nd `  q  1st `  q  2nd `  q  1st `
1211rexbidv 2321 . . 3  q 
Q.  q  2nd `  q  1st `  q 
Q.  q  2nd `  q  1st `
1312opelopab2a 3993 . 2  P.  P.  <. ,  >.  { <. ,  >.  |  P.  P.  q  Q.  q  2nd `  q  1st `  }  q 
Q.  q  2nd `  q  1st `
144, 13syl5bb 181 1  P.  P.  <P  q  Q.  q  2nd `  q  1st `
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242   wcel 1390  wrex 2301   <.cop 3370   class class class wbr 3755   {copab 3808   ` cfv 4845   1stc1st 5707   2ndc2nd 5708   Q.cnq 6264   P.cnp 6275    <P cltp 6279
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-iota 4810  df-fv 4853  df-iltp 6453
This theorem is referenced by:  nqprl  6533  ltprordil  6565  ltpopr  6569  ltsopr  6570  ltaddpr  6571  ltexprlemm  6574  ltexprlemopu  6577  ltexprlemru  6586  aptiprleml  6611  aptiprlemu  6612  archpr  6615  cauappcvgprlem2  6632  caucvgprlem2  6651
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