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Mirrors > Home > MPE Home > Th. List > syl2an23an | Structured version Visualization version GIF version |
Description: Deduction related to syl3an 1156 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 1418 | . 2 ⊢ ((𝜑 ∧ (𝜃 ∧ 𝜑)) → 𝜂) |
6 | 5 | anabss7 671 | 1 ⊢ ((𝜃 ∧ 𝜑) → 𝜂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 |
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 209 df-an 399 df-3an 1085 |
This theorem is referenced by: nf1const 7061 uztrn 12264 ssfzo12bi 13135 modsumfzodifsn 13315 facdiv 13650 swrdnd 14018 cshwidxmod 14167 nndivdvds 15618 pcz 16219 fldivp1 16235 uffix 22531 relogbmul 25357 umgrvad2edg 26997 crctcshwlkn0 27601 satfsschain 32613 satfdm 32618 satffunlem2 32657 |
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