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Mirrors > Home > MPE Home > Th. List > syl2an23an | Structured version Visualization version GIF version |
Description: Deduction related to syl3an 1360 with antecedents in standard conjunction form. (Contributed by Alan Sare, 31-Aug-2016.) |
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.3 | . . 3 ⊢ ((𝜃 ∧ 𝜑) → 𝜏) | |
2 | syl2an23an.1 | . . . 4 ⊢ (𝜑 → 𝜓) | |
3 | syl2an23an.2 | . . . 4 ⊢ (𝜑 → 𝜒) | |
4 | syl2an23an.4 | . . . . 5 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜏) → 𝜂) | |
5 | 4 | 3exp 1256 | . . . 4 ⊢ (𝜓 → (𝜒 → (𝜏 → 𝜂))) |
6 | 2, 3, 5 | sylc 63 | . . 3 ⊢ (𝜑 → (𝜏 → 𝜂)) |
7 | 1, 6 | syl5 33 | . 2 ⊢ (𝜑 → ((𝜃 ∧ 𝜑) → 𝜂)) |
8 | 7 | anabsi7 856 | 1 ⊢ ((𝜃 ∧ 𝜑) → 𝜂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 |
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 196 df-an 385 df-3an 1033 |
This theorem is referenced by: modsumfzodifsn 12605 setsstruct 15727 umgrvad2edg 40440 crctcsh1wlkn0 41024 |
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