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Mirrors > Home > MPE Home > Th. List > syl3an9b | Structured version Visualization version GIF version |
Description: Nested syllogism inference conjoining 3 dissimilar antecedents. (Contributed by NM, 1-May-1995.) |
Ref | Expression |
---|---|
syl3an9b.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
syl3an9b.2 | ⊢ (𝜃 → (𝜒 ↔ 𝜏)) |
syl3an9b.3 | ⊢ (𝜂 → (𝜏 ↔ 𝜁)) |
Ref | Expression |
---|---|
syl3an9b | ⊢ ((𝜑 ∧ 𝜃 ∧ 𝜂) → (𝜓 ↔ 𝜁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl3an9b.1 | . . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | syl3an9b.2 | . . . 4 ⊢ (𝜃 → (𝜒 ↔ 𝜏)) | |
3 | 1, 2 | sylan9bb 509 | . . 3 ⊢ ((𝜑 ∧ 𝜃) → (𝜓 ↔ 𝜏)) |
4 | syl3an9b.3 | . . 3 ⊢ (𝜂 → (𝜏 ↔ 𝜁)) | |
5 | 3, 4 | sylan9bb 509 | . 2 ⊢ (((𝜑 ∧ 𝜃) ∧ 𝜂) → (𝜓 ↔ 𝜁)) |
6 | 5 | 3impa 1108 | 1 ⊢ ((𝜑 ∧ 𝜃 ∧ 𝜂) → (𝜓 ↔ 𝜁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 |
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 396 df-3an 1087 |
This theorem is referenced by: eloprabg 7375 dihjatcclem4 39414 |
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