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Theorem 3anan32 1096
Description: Convert triple conjunction to conjunction, then commute. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
Assertion
Ref Expression
3anan32 ((𝜑𝜓𝜒) ↔ ((𝜑𝜒) ∧ 𝜓))

Proof of Theorem 3anan32
StepHypRef Expression
1 df-3an 1088 . 2 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))
2 an32 646 . 2 (((𝜑𝜓) ∧ 𝜒) ↔ ((𝜑𝜒) ∧ 𝜓))
31, 2bitri 275 1 ((𝜑𝜓𝜒) ↔ ((𝜑𝜒) ∧ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  w3a 1086
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088
This theorem is referenced by:  3ancomb  1098  anandi3r  1102  rabssrabd  4082  dff1o3  6853  bropfvvvvlem  8117  tz7.49c  8487  ispos2  18362  lbsacsbs  21159  obslbs  21751  islbs4  21853  leordtvallem1  23219  trfbas2  23852  isclmp  25131  lssbn  25387  sineq0  26567  dchrelbas3  27283  elno3  27701  nb3grpr2  29401  uspgr2wlkeq  29665  2spthd  29962  clwwlknonwwlknonb  30126  frgr2wwlkeu  30347  elicoelioo  32781  cndprobprob  34441  bnj543  34908  cusgr3cyclex  35142  ellimits  35912  refsymrel2  38569  refsymrel3  38570  dfeqvrel2  38592  dfeqvrel3  38593  i0oii  48824
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