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

Theorem raltpg 3861
 Description: Convert a quantification over a triple to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
Hypotheses
Ref Expression
ralprg.1
ralprg.2
raltpg.3
Assertion
Ref Expression
raltpg
Distinct variable groups:   ,   ,   ,   ,   ,   ,
Allowed substitution hints:   ()   ()   ()   ()

Proof of Theorem raltpg
StepHypRef Expression
1 ralprg.1 . . . . 5
2 ralprg.2 . . . . 5
31, 2ralprg 3859 . . . 4
4 raltpg.3 . . . . 5
54ralsng 3848 . . . 4
63, 5bi2anan9 845 . . 3
763impa 1149 . 2
8 df-tp 3824 . . . 4
98raleqi 2910 . . 3
10 ralunb 3530 . . 3
119, 10bitri 242 . 2
12 df-3an 939 . 2
137, 11, 123bitr4g 281 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360   w3a 937   wceq 1653   wcel 1726  wral 2707   cun 3320  csn 3816  cpr 3817  ctp 3818 This theorem is referenced by:  raltp  3865  nb3grapr  21467  cusgra3v  21478  3v3e3cycl1  21636  constr3trllem2  21643  constr3trllem5  21646  f13dfv  28099  frgra3v  28466 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-v 2960  df-sbc 3164  df-un 3327  df-sn 3822  df-pr 3823  df-tp 3824
 Copyright terms: Public domain W3C validator