Theorem syl2an23an 1451

Description: Deduction related to syl3an 1178 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 1450	. 2 ⊢ ((𝜑 ∧ (𝜃 ∧ 𝜑)) → 𝜂)
6	5	anabss7 686	1 ⊢ ((𝜃 ∧ 𝜑) → 𝜂)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105

This theorem is used by:  nf1const  7335  uztrn  13010  ssfzo12bi  13923  modsumfzodifsn  14114  facdiv  14458  swrdnd  14832  cshwidxmod  14982  nndivdvds  16460  pcz  17088  fldivp1  17104  uffix  24270  relogbmul  27135  umgrvad2edg  29824  crctcshwlkn0  30440  satfsschain  36296  satfdm  36301  satffunlem2  36340  modmkpkne  48642  pgn4cyclex  49429