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Mirrors > Home > MPE Home > Th. List > 3adant2l | Structured version Visualization version GIF version |
Description: Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.) (Proof shortened by Wolf Lammen, 25-Jun-2022.) |
Ref | Expression |
---|---|
ad4ant3.1 | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) |
Ref | Expression |
---|---|
3adant2l | ⊢ ((𝜑 ∧ (𝜏 ∧ 𝜓) ∧ 𝜒) → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 487 | . 2 ⊢ ((𝜏 ∧ 𝜓) → 𝜓) | |
2 | ad4ant3.1 | . 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | |
3 | 1, 2 | syl3an2 1160 | 1 ⊢ ((𝜑 ∧ (𝜏 ∧ 𝜓) ∧ 𝜒) → 𝜃) |
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: axdc3lem4 9875 modexp 13600 lmmbr2 23862 ax5seglem1 26714 ax5seglem2 26715 nvaddsub4 28434 pl1cn 31198 athgt 36607 ltrncoelN 37294 ltrncoat 37295 trlcoabs 37872 tendoplcl2 37929 tendopltp 37931 dih1dimatlem0 38479 pellex 39452 fourierdlem42 42454 |
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