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Mirrors > Home > ILE Home > Th. List > sylanbr | GIF version |
Description: A syllogism inference. (Contributed by NM, 18-May-1994.) |
Ref | Expression |
---|---|
sylanbr.1 | ⊢ (𝜓 ↔ 𝜑) |
sylanbr.2 | ⊢ ((𝜓 ∧ 𝜒) → 𝜃) |
Ref | Expression |
---|---|
sylanbr | ⊢ ((𝜑 ∧ 𝜒) → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sylanbr.1 | . . 3 ⊢ (𝜓 ↔ 𝜑) | |
2 | 1 | biimpri 124 | . 2 ⊢ (𝜑 → 𝜓) |
3 | sylanbr.2 | . 2 ⊢ ((𝜓 ∧ 𝜒) → 𝜃) | |
4 | 2, 3 | sylan 267 | 1 ⊢ ((𝜑 ∧ 𝜒) → 𝜃) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: syl2anbr 276 mosubt 2718 xpiindim 4473 funfvdm 5236 caovimo 5694 tfrlem7 5933 iinerm 6178 expclzaplem 9279 expgt0 9288 expge0 9291 expge1 9292 |
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