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Mirrors > Home > ILE Home > Th. List > syl2and | GIF version |
Description: A syllogism deduction. (Contributed by NM, 15-Dec-2004.) |
Ref | Expression |
---|---|
syl2and.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
syl2and.2 | ⊢ (𝜑 → (𝜃 → 𝜏)) |
syl2and.3 | ⊢ (𝜑 → ((𝜒 ∧ 𝜏) → 𝜂)) |
Ref | Expression |
---|---|
syl2and | ⊢ (𝜑 → ((𝜓 ∧ 𝜃) → 𝜂)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl2and.1 | . 2 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
2 | syl2and.2 | . . 3 ⊢ (𝜑 → (𝜃 → 𝜏)) | |
3 | syl2and.3 | . . 3 ⊢ (𝜑 → ((𝜒 ∧ 𝜏) → 𝜂)) | |
4 | 2, 3 | sylan2d 292 | . 2 ⊢ (𝜑 → ((𝜒 ∧ 𝜃) → 𝜂)) |
5 | 1, 4 | syland 291 | 1 ⊢ (𝜑 → ((𝜓 ∧ 𝜃) → 𝜂)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: anim12d 333 recexprlem1ssl 7566 recexprlem1ssu 7567 xle2add 9807 fzen 9969 bezoutlembi 11927 rpmulgcd2 12016 pcqmul 12224 |
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