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Mirrors > Home > MPE Home > Th. List > imdi | Structured version Visualization version GIF version |
Description: Distributive law for implication. Compare Theorem *5.41 of [WhiteheadRussell] p. 125. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
imdi | ⊢ ((𝜑 → (𝜓 → 𝜒)) ↔ ((𝜑 → 𝜓) → (𝜑 → 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-2 7 | . 2 ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒))) | |
2 | pm2.86 109 | . 2 ⊢ (((𝜑 → 𝜓) → (𝜑 → 𝜒)) → (𝜑 → (𝜓 → 𝜒))) | |
3 | 1, 2 | impbii 212 | 1 ⊢ ((𝜑 → (𝜓 → 𝜒)) ↔ ((𝜑 → 𝜓) → (𝜑 → 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 |
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 |
This theorem is referenced by: pm5.41 395 orimdi 931 bnj1174 32696 ifpim23g 40787 |
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