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Mirrors > Home > ILE Home > Th. List > syl3anb | GIF version |
Description: A triple syllogism inference. (Contributed by NM, 15-Oct-2005.) |
Ref | Expression |
---|---|
syl3anb.1 | ⊢ (𝜑 ↔ 𝜓) |
syl3anb.2 | ⊢ (𝜒 ↔ 𝜃) |
syl3anb.3 | ⊢ (𝜏 ↔ 𝜂) |
syl3anb.4 | ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜂) → 𝜁) |
Ref | Expression |
---|---|
syl3anb | ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl3anb.1 | . . 3 ⊢ (𝜑 ↔ 𝜓) | |
2 | syl3anb.2 | . . 3 ⊢ (𝜒 ↔ 𝜃) | |
3 | syl3anb.3 | . . 3 ⊢ (𝜏 ↔ 𝜂) | |
4 | 1, 2, 3 | 3anbi123i 1183 | . 2 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) ↔ (𝜓 ∧ 𝜃 ∧ 𝜂)) |
5 | syl3anb.4 | . 2 ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜂) → 𝜁) | |
6 | 4, 5 | sylbi 120 | 1 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜁) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∧ w3a 973 |
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 df-3an 975 |
This theorem is referenced by: syl3anbr 1277 poxp 6211 |
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