Theorem syl2an23an 1423

Description: Deduction related to syl3an 1160 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 1422	. 2 ⊢ ((𝜑 ∧ (𝜃 ∧ 𝜑)) → 𝜂)
6	5	anabss7 672	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  7340  uztrn  12921  ssfzo12bi  13811  modsumfzodifsn  13995  facdiv  14336  swrdnd  14702  cshwidxmod  14851  nndivdvds  16311  pcz  16928  fldivp1  16944  uffix  23950  relogbmul  26838  umgrvad2edg  29248  crctcshwlkn0  29854  satfsschain  35332  satfdm  35337  satffunlem2  35376