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

Theorem rextpg 3852
 Description: Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.)
Hypotheses
Ref Expression
ralprg.1
ralprg.2
raltpg.3
Assertion
Ref Expression
rextpg
Distinct variable groups:   ,   ,   ,   ,   ,   ,
Allowed substitution hints:   ()   ()   ()   ()

Proof of Theorem rextpg
StepHypRef Expression
1 ralprg.1 . . . . . 6
2 ralprg.2 . . . . . 6
31, 2rexprg 3850 . . . . 5
43orbi1d 684 . . . 4
5 raltpg.3 . . . . . 6
65rexsng 3839 . . . . 5
76orbi2d 683 . . . 4
84, 7sylan9bb 681 . . 3
983impa 1148 . 2
10 df-tp 3814 . . . 4
1110rexeqi 2901 . . 3
12 rexun 3519 . . 3
1311, 12bitri 241 . 2
14 df-3or 937 . 2
159, 13, 143bitr4g 280 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wo 358   wa 359   w3o 935   w3a 936   wceq 1652   wcel 1725  wrex 2698   cun 3310  csn 3806  cpr 3807  ctp 3808 This theorem is referenced by:  rextp  3856  fr3nr  4752  nb3graprlem2  21453  frgra3vlem2  28328  3vfriswmgra  28332 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rex 2703  df-v 2950  df-sbc 3154  df-un 3317  df-sn 3812  df-pr 3813  df-tp 3814
 Copyright terms: Public domain W3C validator