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Mirrors > Home > MPE Home > Th. List > Mathboxes > bi13impia | Structured version Visualization version GIF version |
Description: 3impia 1115 with the outer implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.) |
Ref | Expression |
---|---|
bi13impia.1 | ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜒 → 𝜃)) |
Ref | Expression |
---|---|
bi13impia | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bi13impia.1 | . . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜒 → 𝜃)) | |
2 | 1 | biimpi 215 | . 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃)) |
3 | 2 | 3impia 1115 | 1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) |
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: (None) |
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