Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ralpr Structured version   Unicode version

Theorem ralpr 4086
 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
ralpr.2
ralpr.3
ralpr.4
Assertion
Ref Expression
ralpr
Distinct variable groups:   ,   ,   ,   ,
Allowed substitution hint:   ()

Proof of Theorem ralpr
StepHypRef Expression
1 ralpr.1 . 2
2 ralpr.2 . 2
3 ralpr.3 . . 3
4 ralpr.4 . . 3
53, 4ralprg 4082 . 2
61, 2, 5mp2an 672 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wceq 1379   wcel 1767  wral 2817  cvv 3118  cpr 4035 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2822  df-v 3120  df-sbc 3337  df-un 3486  df-sn 4034  df-pr 4036 This theorem is referenced by:  fzprval  11752  xpsfrnel  14835  xpsle  14853  isdrs2  15443  pmtrsn  16417  iblcnlem1  22062  wlkntrllem2  24385  wlkntrllem3  24386  2wlklem  24389  usgra2wlkspthlem1  24442  numclwwlkovf2ex  24910  subfacp1lem3  28451  fprb  29130  usgra2pthspth  32141  ldepsnlinc  32591
 Copyright terms: Public domain W3C validator