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Mirrors > Home > ILE Home > Th. List > syld3an1 | GIF version |
Description: A syllogism inference. (Contributed by NM, 7-Jul-2008.) |
Ref | Expression |
---|---|
syld3an1.1 | ⊢ ((𝜒 ∧ 𝜓 ∧ 𝜃) → 𝜑) |
syld3an1.2 | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏) |
Ref | Expression |
---|---|
syld3an1 | ⊢ ((𝜒 ∧ 𝜓 ∧ 𝜃) → 𝜏) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syld3an1.1 | . . . 4 ⊢ ((𝜒 ∧ 𝜓 ∧ 𝜃) → 𝜑) | |
2 | 1 | 3com13 1109 | . . 3 ⊢ ((𝜃 ∧ 𝜓 ∧ 𝜒) → 𝜑) |
3 | syld3an1.2 | . . . 4 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏) | |
4 | 3 | 3com13 1109 | . . 3 ⊢ ((𝜃 ∧ 𝜓 ∧ 𝜑) → 𝜏) |
5 | 2, 4 | syld3an3 1180 | . 2 ⊢ ((𝜃 ∧ 𝜓 ∧ 𝜒) → 𝜏) |
6 | 5 | 3com13 1109 | 1 ⊢ ((𝜒 ∧ 𝜓 ∧ 𝜃) → 𝜏) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 885 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 |
This theorem depends on definitions: df-bi 110 df-3an 887 |
This theorem is referenced by: npncan 7232 nnpcan 7234 ppncan 7253 div2negap 7711 ltmuldiv 7840 mulqmod0 9172 |
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