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Mirrors > Home > MPE Home > Th. List > Mathboxes > 3impcombi | Structured version Visualization version GIF version |
Description: A 1-hypothesis propositional calculus deduction. (Contributed by Alan Sare, 25-Sep-2017.) |
Ref | Expression |
---|---|
3impcombi.1 | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜑) → (𝜒 ↔ 𝜃)) |
Ref | Expression |
---|---|
3impcombi | ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜒) → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3impcombi.1 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜑) → (𝜒 ↔ 𝜃)) | |
2 | 1 | biimpd 232 | . . . 4 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜑) → (𝜒 → 𝜃)) |
3 | 2 | 3anidm13 1422 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃)) |
4 | 3 | ancoms 462 | . 2 ⊢ ((𝜓 ∧ 𝜑) → (𝜒 → 𝜃)) |
5 | 4 | 3impia 1119 | 1 ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜒) → 𝜃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ w3a 1089 |
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 1091 |
This theorem is referenced by: isosctrlem1ALT 42168 |
Copyright terms: Public domain | W3C validator |