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	⊢ (𝜑 → 𝜓)
syl2an23an.2	⊢ (𝜑 → 𝜒)
syl2an23an.3	⊢ ((𝜃 ∧ 𝜑) → 𝜏)
syl2an23an.4	⊢ ((𝜓 ∧ 𝜒 ∧ 𝜏) → 𝜂)

Assertion

Ref	Expression
syl2an23an	⊢ ((𝜃 ∧ 𝜑) → 𝜂)

Proof of Theorem syl2an23an

Step	Hyp	Ref	Expression
1		syl2an23an.1	. . . 4 ⊢ (𝜑 → 𝜓)
2		syl2an23an.2	. . . 4 ⊢ (𝜑 → 𝜒)
3		syl2an23an.3	. . . 4 ⊢ ((𝜃 ∧ 𝜑) → 𝜏)
4		syl2an23an.4	. . . 4 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜏) → 𝜂)
5	1, 2, 3, 4	syl2an3an 1533	. . 3 ⊢ ((𝜑 ∧ (𝜃 ∧ 𝜑)) → 𝜂)
6	5	ex 449	. 2 ⊢ (𝜑 → ((𝜃 ∧ 𝜑) → 𝜂))
7	6	anabsi7 895	1 ⊢ ((𝜃 ∧ 𝜑) → 𝜂)

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