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Mirrors > Home > MPE Home > Th. List > Mathboxes > hirstL-ax3 | Structured version Visualization version GIF version |
Description: The third axiom of a system called "L" but proven to be a theorem since set.mm uses a different third axiom. This is named hirst after Holly P. Hirst and Jeffry L. Hirst. Axiom A3 of [Mendelson] p. 35. (Contributed by Jarvin Udandy, 7-Feb-2015.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
hirstL-ax3 | ⊢ ((¬ 𝜑 → ¬ 𝜓) → ((¬ 𝜑 → 𝜓) → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm4.64 846 | . 2 ⊢ ((¬ 𝜑 → 𝜓) ↔ (𝜑 ∨ 𝜓)) | |
2 | pm4.66 847 | . . 3 ⊢ ((¬ 𝜑 → ¬ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓)) | |
3 | pm2.64 939 | . . . 4 ⊢ ((𝜑 ∨ 𝜓) → ((𝜑 ∨ ¬ 𝜓) → 𝜑)) | |
4 | 3 | com12 32 | . . 3 ⊢ ((𝜑 ∨ ¬ 𝜓) → ((𝜑 ∨ 𝜓) → 𝜑)) |
5 | 2, 4 | sylbi 216 | . 2 ⊢ ((¬ 𝜑 → ¬ 𝜓) → ((𝜑 ∨ 𝜓) → 𝜑)) |
6 | 1, 5 | syl5bi 241 | 1 ⊢ ((¬ 𝜑 → ¬ 𝜓) → ((¬ 𝜑 → 𝜓) → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 844 |
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-or 845 |
This theorem is referenced by: ax3h 44356 |
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