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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 417 | . . 3 ⊢ (𝜑 → (𝜒 → (𝜃 → 𝜏))) |
4 | 1, 3 | syld 47 | . 2 ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜏))) |
5 | 4 | impd 412 | 1 ⊢ (𝜑 → ((𝜓 ∧ 𝜃) → 𝜏)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 |
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 206 df-an 398 |
This theorem is referenced by: sylani 605 sylan2d 606 syl2and 609 onfununi 8335 lt2add 11694 nn0seqcvgd 16502 1stcelcls 22946 llyidm 22973 filuni 23370 ballotlemimin 33441 btwnintr 34928 ifscgr 34953 btwnconn1lem12 35007 poimir 36458 cvrntr 38233 goldbachthlem2 46148 |
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