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Theorem syl2an23an 1231
Description: Deduction related to syl3an 1212 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 1138 . . . 4 (𝜓 → (𝜒 → (𝜏𝜂)))
62, 3, 5sylc 61 . . 3 (𝜑 → (𝜏𝜂))
71, 6syl5 32 . 2 (𝜑 → ((𝜃𝜑) → 𝜂))
87anabsi7 546 1 ((𝜃𝜑) → 𝜂)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  w3a 920
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106
This theorem depends on definitions:  df-bi 115  df-3an 922
This theorem is referenced by: (None)
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