Theorem syl2an23an 1422

Description: Deduction related to syl3an 1159 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 1421	. 2 ⊢ ((𝜑 ∧ (𝜃 ∧ 𝜑)) → 𝜂)
6	5	anabss7 670	1 ⊢ ((𝜃 ∧ 𝜑) → 𝜂)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086

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 397  df-3an 1088

This theorem is referenced by:  nf1const  7178  uztrn  12598  ssfzo12bi  13480  modsumfzodifsn  13662  facdiv  13999  swrdnd  14365  cshwidxmod  14514  nndivdvds  15970  pcz  16580  fldivp1  16596  uffix  23070  relogbmul  25925  umgrvad2edg  27578  crctcshwlkn0  28183  satfsschain  33323  satfdm  33328  satffunlem2  33367