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Mirrors > Home > MPE Home > Th. List > syland | Structured version Visualization version GIF version |
Description: A syllogism deduction. (Contributed by NM, 15-Dec-2004.) |
Ref | Expression |
---|---|
syland.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
syland.2 | ⊢ (𝜑 → ((𝜒 ∧ 𝜃) → 𝜏)) |
Ref | Expression |
---|---|
syland | ⊢ (𝜑 → ((𝜓 ∧ 𝜃) → 𝜏)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syland.1 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
2 | syland.2 | . . . 4 ⊢ (𝜑 → ((𝜒 ∧ 𝜃) → 𝜏)) | |
3 | 2 | expd 451 | . . 3 ⊢ (𝜑 → (𝜒 → (𝜃 → 𝜏))) |
4 | 1, 3 | syld 46 | . 2 ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜏))) |
5 | 4 | impd 446 | 1 ⊢ (𝜑 → ((𝜓 ∧ 𝜃) → 𝜏)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 |
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 |
This theorem is referenced by: sylan2d 498 syl2and 499 sylani 684 onfununi 7325 lt2add 10392 nn0seqcvgd 15121 1stcelcls 21074 llyidm 21101 filuni 21499 ballotlemimin 29894 btwnintr 31296 ifscgr 31321 btwnconn1lem12 31375 poimir 32612 cvrntr 33729 goldbachthlem2 39996 |
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