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Mirrors > Home > MPE Home > Th. List > Mathboxes > bi123imp0 | Structured version Visualization version GIF version |
Description: Similar to 3imp 1108 except all implications are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.) |
Ref | Expression |
---|---|
bi23imp0.1 | ⊢ (𝜑 ↔ (𝜓 ↔ (𝜒 ↔ 𝜃))) |
Ref | Expression |
---|---|
bi123imp0 | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bi23imp0.1 | . . 3 ⊢ (𝜑 ↔ (𝜓 ↔ (𝜒 ↔ 𝜃))) | |
2 | biimp 218 | . . . 4 ⊢ ((𝜓 ↔ (𝜒 ↔ 𝜃)) → (𝜓 → (𝜒 ↔ 𝜃))) | |
3 | biimp 218 | . . . 4 ⊢ ((𝜒 ↔ 𝜃) → (𝜒 → 𝜃)) | |
4 | 2, 3 | syl6 35 | . . 3 ⊢ ((𝜓 ↔ (𝜒 ↔ 𝜃)) → (𝜓 → (𝜒 → 𝜃))) |
5 | 1, 4 | sylbi 220 | . 2 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) |
6 | 5 | 3imp 1108 | 1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ w3a 1084 |
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 210 df-an 400 df-3an 1086 |
This theorem is referenced by: (None) |
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