Theorem syl2an23an 1421

Description: Deduction related to syl3an 1158 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	. . 3 ⊢ (𝜑 → 𝜓)
2		syl2an23an.2	. . 3 ⊢ (𝜑 → 𝜒)
3		syl2an23an.3	. . 3 ⊢ ((𝜃 ∧ 𝜑) → 𝜏)
4		syl2an23an.4	. . 3 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜏) → 𝜂)
5	1, 2, 3, 4	syl2an3an 1420	. 2 ⊢ ((𝜑 ∧ (𝜃 ∧ 𝜑)) → 𝜂)
6	5	anabss7 669	1 ⊢ ((𝜃 ∧ 𝜑) → 𝜂)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085

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 396  df-3an 1087

This theorem is referenced by:  nf1const  7156  uztrn  12529  ssfzo12bi  13410  modsumfzodifsn  13592  facdiv  13929  swrdnd  14295  cshwidxmod  14444  nndivdvds  15900  pcz  16510  fldivp1  16526  uffix  22980  relogbmul  25832  umgrvad2edg  27483  crctcshwlkn0  28087  satfsschain  33226  satfdm  33231  satffunlem2  33270