Theorem syl2an23an 1426

Description: Deduction related to syl3an 1161 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 1425	. 2 ⊢ ((𝜑 ∧ (𝜃 ∧ 𝜑)) → 𝜂)
6	5	anabss7 674	1 ⊢ ((𝜃 ∧ 𝜑) → 𝜂)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087

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 207  df-an 396  df-3an 1089

This theorem is referenced by:  nf1const  7260  uztrn  12781  ssfzo12bi  13689  modsumfzodifsn  13879  facdiv  14222  swrdnd  14590  cshwidxmod  14738  nndivdvds  16200  pcz  16821  fldivp1  16837  uffix  23877  relogbmul  26755  umgrvad2edg  29298  crctcshwlkn0  29906  satfsschain  35577  satfdm  35582  satffunlem2  35621  modmkpkne  47715  pgn4cyclex  48480