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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-peirce | Structured version Visualization version GIF version |
Description: Proof of peirce 193 from minimal implicational calculus, the axiomatic definition of disjunction (olc 399, orc 400, jao 534), and Curry's axiom bj-curry 32542. (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bj-peirce | ⊢ (((𝜑 → 𝜓) → 𝜑) → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-curry 32542 | . . 3 ⊢ (𝜑 ∨ (𝜑 → 𝜓)) | |
2 | bj-orim2 32541 | . . 3 ⊢ (((𝜑 → 𝜓) → 𝜑) → ((𝜑 ∨ (𝜑 → 𝜓)) → (𝜑 ∨ 𝜑))) | |
3 | 1, 2 | mpi 20 | . 2 ⊢ (((𝜑 → 𝜓) → 𝜑) → (𝜑 ∨ 𝜑)) |
4 | pm1.2 535 | . 2 ⊢ ((𝜑 ∨ 𝜑) → 𝜑) | |
5 | 3, 4 | syl 17 | 1 ⊢ (((𝜑 → 𝜓) → 𝜑) → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 383 |
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 197 df-or 385 df-an 386 |
This theorem is referenced by: (None) |
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