![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > sylancb | GIF version |
Description: A syllogism inference combined with contraction. (Contributed by NM, 3-Sep-2004.) |
Ref | Expression |
---|---|
sylancb.1 | ⊢ (𝜑 ↔ 𝜓) |
sylancb.2 | ⊢ (𝜑 ↔ 𝜒) |
sylancb.3 | ⊢ ((𝜓 ∧ 𝜒) → 𝜃) |
Ref | Expression |
---|---|
sylancb | ⊢ (𝜑 → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sylancb.1 | . . 3 ⊢ (𝜑 ↔ 𝜓) | |
2 | sylancb.2 | . . 3 ⊢ (𝜑 ↔ 𝜒) | |
3 | sylancb.3 | . . 3 ⊢ ((𝜓 ∧ 𝜒) → 𝜃) | |
4 | 1, 2, 3 | syl2anb 291 | . 2 ⊢ ((𝜑 ∧ 𝜑) → 𝜃) |
5 | 4 | anidms 397 | 1 ⊢ (𝜑 → 𝜃) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 |
This theorem depends on definitions: df-bi 117 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |