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Theorem syl2an23an 1534
Description: Deduction related to syl3an 1164 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 . . . 4 (𝜑𝜓)
2 syl2an23an.2 . . . 4 (𝜑𝜒)
3 syl2an23an.3 . . . 4 ((𝜃𝜑) → 𝜏)
4 syl2an23an.4 . . . 4 ((𝜓𝜒𝜏) → 𝜂)
51, 2, 3, 4syl2an3an 1533 . . 3 ((𝜑 ∧ (𝜃𝜑)) → 𝜂)
65ex 449 . 2 (𝜑 → ((𝜃𝜑) → 𝜂))
76anabsi7 895 1 ((𝜃𝜑) → 𝜂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072
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 385  df-3an 1074
This theorem is referenced by:  modsumfzodifsn  12937  setsstructOLD  16101  umgrvad2edg  26304  crctcshwlkn0  26924  clwwlknonwwlknonbOLD  27255
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