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Theorem rextpg 4079
 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 4077 . . . . 5
43orbi1d 702 . . . 4
5 raltpg.3 . . . . . 6
65rexsng 4063 . . . . 5
76orbi2d 701 . . . 4
84, 7sylan9bb 699 . . 3
983impa 1191 . 2
10 df-tp 4032 . . . 4
1110rexeqi 3063 . . 3
12 rexun 3684 . . 3
1311, 12bitri 249 . 2
14 df-3or 974 . 2
159, 13, 143bitr4g 288 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wo 368   wa 369   w3o 972   w3a 973   wceq 1379   wcel 1767  wrex 2815   cun 3474  csn 4027  cpr 4029  ctp 4031 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-3or 974  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-rex 2820  df-v 3115  df-sbc 3332  df-un 3481  df-sn 4028  df-pr 4030  df-tp 4032 This theorem is referenced by:  rextp  4083  fr3nr  6599  nb3graprlem2  24156  frgra3vlem2  24705  3vfriswmgra  24709
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