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Theorem ralpr 3416
Description: Convert a quantification over a pair to a conjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.)
Hypotheses
Ref Expression
ralpr.1  _V
ralpr.2  _V
ralpr.3
ralpr.4
Assertion
Ref Expression
ralpr  { ,  }
Distinct variable groups:   ,   ,   ,   ,
Allowed substitution hint:   ()

Proof of Theorem ralpr
StepHypRef Expression
1 ralpr.1 . 2  _V
2 ralpr.2 . 2  _V
3 ralpr.3 . . 3
4 ralpr.4 . . 3
53, 4ralprg 3412 . 2  _V  _V 
{ ,  }
61, 2, 5mp2an 402 1  { ,  }
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242   wcel 1390  wral 2300   _Vcvv 2551   {cpr 3368
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-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-sbc 2759  df-un 2916  df-sn 3373  df-pr 3374
This theorem is referenced by:  fzprval  8714
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