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Mirrors > Home > ILE Home > Th. List > sylsyld | GIF version |
Description: A double syllogism inference. (Contributed by Alan Sare, 20-Apr-2011.) |
Ref | Expression |
---|---|
sylsyld.1 | ⊢ (𝜑 → 𝜓) |
sylsyld.2 | ⊢ (𝜑 → (𝜒 → 𝜃)) |
sylsyld.3 | ⊢ (𝜓 → (𝜃 → 𝜏)) |
Ref | Expression |
---|---|
sylsyld | ⊢ (𝜑 → (𝜒 → 𝜏)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sylsyld.2 | . 2 ⊢ (𝜑 → (𝜒 → 𝜃)) | |
2 | sylsyld.1 | . . 3 ⊢ (𝜑 → 𝜓) | |
3 | sylsyld.3 | . . 3 ⊢ (𝜓 → (𝜃 → 𝜏)) | |
4 | 2, 3 | syl 14 | . 2 ⊢ (𝜑 → (𝜃 → 𝜏)) |
5 | 1, 4 | syld 45 | 1 ⊢ (𝜑 → (𝜒 → 𝜏)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 |
This theorem is referenced by: ax10o 1651 a16g 1793 trintssm 3958 funimaexglem 5110 smoiun 6080 findcard2 6659 ltexprlemrl 7223 archsr 7381 elfz0ubfz0 9590 |
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