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Theorem rextpg 3778
 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 3776 . . . . 5
43orbi1d 683 . . . 4
5 raltpg.3 . . . . . 6
65rexsng 3766 . . . . 5
76orbi2d 682 . . . 4
84, 7sylan9bb 680 . . 3
983impa 1146 . 2
10 df-tp 3743 . . . 4
1110rexeqi 2812 . . 3
12 rexun 3443 . . 3
1311, 12bitri 240 . 2
14 df-3or 935 . 2
159, 13, 143bitr4g 279 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 176   wo 357   wa 358   w3o 933   w3a 934   wceq 1642   wcel 1710  wrex 2615   cun 3207  csn 3737  cpr 3738  ctp 3739 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742  df-tp 3743 This theorem is referenced by:  rextp  3782
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