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Mirrors > Home > MPE Home > Th. List > syl2an3an | Structured version Visualization version GIF version |
Description: syl3an 1368 with antecedents in standard conjunction form. (Contributed by Alan Sare, 31-Aug-2016.) |
Ref | Expression |
---|---|
syl2an3an.1 | ⊢ (𝜑 → 𝜓) |
syl2an3an.2 | ⊢ (𝜑 → 𝜒) |
syl2an3an.3 | ⊢ (𝜃 → 𝜏) |
syl2an3an.4 | ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜏) → 𝜂) |
Ref | Expression |
---|---|
syl2an3an | ⊢ ((𝜑 ∧ 𝜃) → 𝜂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl2an3an.1 | . . 3 ⊢ (𝜑 → 𝜓) | |
2 | syl2an3an.2 | . . 3 ⊢ (𝜑 → 𝜒) | |
3 | syl2an3an.3 | . . 3 ⊢ (𝜃 → 𝜏) | |
4 | syl2an3an.4 | . . 3 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜏) → 𝜂) | |
5 | 1, 2, 3, 4 | syl3an 1368 | . 2 ⊢ ((𝜑 ∧ 𝜑 ∧ 𝜃) → 𝜂) |
6 | 5 | 3anidm12 1383 | 1 ⊢ ((𝜑 ∧ 𝜃) → 𝜂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 |
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 197 df-an 386 df-3an 1039 |
This theorem is referenced by: disjxiun 4649 funcnvtp 5951 funcnvqpOLD 5953 cncongr1 15381 gausslemma2dlem2 25092 eucrctshift 27103 extwwlkfab 27223 fmtnofac2lem 41480 |
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