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Mirrors > Home > MPE Home > Th. List > syl2an23an | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
syl2an23an.1 | ⊢ (𝜑 → 𝜓) |
syl2an23an.2 | ⊢ (𝜑 → 𝜒) |
syl2an23an.3 | ⊢ ((𝜃 ∧ 𝜑) → 𝜏) |
syl2an23an.4 | ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜏) → 𝜂) |
Ref | Expression |
---|---|
syl2an23an | ⊢ ((𝜃 ∧ 𝜑) → 𝜂) |
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 673 | 1 ⊢ ((𝜃 ∧ 𝜑) → 𝜂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ 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 207 df-an 396 df-3an 1088 |
This theorem is referenced by: nf1const 7324 uztrn 12894 ssfzo12bi 13797 modsumfzodifsn 13982 facdiv 14323 swrdnd 14689 cshwidxmod 14838 nndivdvds 16296 pcz 16915 fldivp1 16931 uffix 23945 relogbmul 26835 umgrvad2edg 29245 crctcshwlkn0 29851 satfsschain 35349 satfdm 35354 satffunlem2 35393 |
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