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Theorem syl2an23an 1384
 Description: Deduction related to syl3an 1365 with antecedents in standard conjunction form. (Contributed by Alan Sare, 31-Aug-2016.)
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.3 . . 3 ((𝜃𝜑) → 𝜏)
2 syl2an23an.1 . . . 4 (𝜑𝜓)
3 syl2an23an.2 . . . 4 (𝜑𝜒)
4 syl2an23an.4 . . . . 5 ((𝜓𝜒𝜏) → 𝜂)
543exp 1261 . . . 4 (𝜓 → (𝜒 → (𝜏𝜂)))
62, 3, 5sylc 65 . . 3 (𝜑 → (𝜏𝜂))
71, 6syl5 34 . 2 (𝜑 → ((𝜃𝜑) → 𝜂))
87anabsi7 859 1 ((𝜃𝜑) → 𝜂)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1036 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 197  df-an 386  df-3an 1038 This theorem is referenced by:  modsumfzodifsn  12691  setsstructOLD  15831  umgrvad2edg  26015  crctcshwlkn0  26599
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