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Theorem raltpg 4025
 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 4023 . . . 4
4 raltpg.3 . . . . 5
54ralsng 4008 . . . 4
63, 5bi2anan9 885 . . 3
763impa 1204 . 2
8 df-tp 3975 . . . 4
98raleqi 2993 . . 3
10 ralunb 3617 . . 3
119, 10bitri 253 . 2
12 df-3an 988 . 2
137, 11, 123bitr4g 292 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 188   wa 371   w3a 986   wceq 1446   wcel 1889  wral 2739   cun 3404  csn 3970  cpr 3972  ctp 3974 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ral 2744  df-v 3049  df-sbc 3270  df-un 3411  df-sn 3971  df-pr 3973  df-tp 3975 This theorem is referenced by:  raltp  4029  raltpd  4098  f13dfv  6178  sumtp  13822  lcmftp  14621  nb3grapr  25193  cusgra3v  25204  3v3e3cycl1  25384  constr3trllem2  25391  constr3trllem5  25394  frgra3v  25742  nb3grpr  39466
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