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Theorem raltpg 4080
 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:   ,   ,   ,   ,   ,   ,

Proof of Theorem raltpg
StepHypRef Expression
1 ralprg.1 . . . . 5
2 ralprg.2 . . . . 5
31, 2ralprg 4078 . . . 4
4 raltpg.3 . . . . 5
54ralsng 4064 . . . 4
63, 5bi2anan9 873 . . 3
763impa 1191 . 2
8 df-tp 4034 . . . 4
98raleqi 3058 . . 3
10 ralunb 3684 . . 3
119, 10bitri 249 . 2
12 df-3an 975 . 2
137, 11, 123bitr4g 288 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  <->wb 184  /\wa 369  /\w3a 973  =wceq 1395  e.wcel 1818  A.wral 2807  u.cun 3473  {csn 4029  {cpr 4031  {ctp 4033 This theorem is referenced by:  raltp  4084  raltpd  4153  f13dfv  6180  nb3grapr  24453  cusgra3v  24464  3v3e3cycl1  24644  constr3trllem2  24651  constr3trllem5  24654  frgra3v  25002 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-v 3111  df-sbc 3328  df-un 3480  df-sn 4030  df-pr 4032  df-tp 4034
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