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Theorem syl2an23an 1423
Description: Deduction related to syl3an 1160 with antecedents in standard conjunction form. (Contributed by Alan Sare, 31-Aug-2016.) (Proof shortened by Wolf Lammen, 28-Jun-2022.)
Hypotheses
Ref Expression
syl2an23an.1 (𝜑𝜓)
syl2an23an.2 (𝜑𝜒)
syl2an23an.3 ((𝜃𝜑) → 𝜏)
syl2an23an.4 ((𝜓𝜒𝜏) → 𝜂)
Assertion
Ref Expression
syl2an23an ((𝜃𝜑) → 𝜂)

Proof of Theorem syl2an23an
StepHypRef Expression
1 syl2an23an.1 . . 3 (𝜑𝜓)
2 syl2an23an.2 . . 3 (𝜑𝜒)
3 syl2an23an.3 . . 3 ((𝜃𝜑) → 𝜏)
4 syl2an23an.4 . . 3 ((𝜓𝜒𝜏) → 𝜂)
51, 2, 3, 4syl2an3an 1422 . 2 ((𝜑 ∧ (𝜃𝜑)) → 𝜂)
65anabss7 671 1 ((𝜃𝜑) → 𝜂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087
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 206  df-an 397  df-3an 1089
This theorem is referenced by:  nf1const  7301  uztrn  12839  ssfzo12bi  13726  modsumfzodifsn  13908  facdiv  14246  swrdnd  14603  cshwidxmod  14752  nndivdvds  16205  pcz  16813  fldivp1  16829  uffix  23424  relogbmul  26279  umgrvad2edg  28467  crctcshwlkn0  29072  satfsschain  34350  satfdm  34355  satffunlem2  34394
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