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Mirrors > Home > MPE Home > Th. List > re2luk3 | Structured version Visualization version GIF version |
Description: luk-3 1665 derived from Russell-Bernays'.
This theorem, along with re1axmp 1772, re2luk1 1773, and re2luk2 1774 shows that rb-ax1 1760, rb-ax2 1761, rb-ax3 1762, and rb-ax4 1763, along with anmp 1759, can be used as a complete axiomatization of propositional calculus. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
re2luk3 | ⊢ (𝜑 → (¬ 𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rb-imdf 1758 | . . . 4 ⊢ ¬ (¬ (¬ (¬ 𝜑 → 𝜓) ∨ (¬ ¬ 𝜑 ∨ 𝜓)) ∨ ¬ (¬ (¬ ¬ 𝜑 ∨ 𝜓) ∨ (¬ 𝜑 → 𝜓))) | |
2 | 1 | rblem7 1771 | . . 3 ⊢ (¬ (¬ ¬ 𝜑 ∨ 𝜓) ∨ (¬ 𝜑 → 𝜓)) |
3 | rb-ax4 1763 | . . . . . 6 ⊢ (¬ (¬ 𝜑 ∨ ¬ 𝜑) ∨ ¬ 𝜑) | |
4 | rb-ax3 1762 | . . . . . 6 ⊢ (¬ ¬ 𝜑 ∨ (¬ 𝜑 ∨ ¬ 𝜑)) | |
5 | 3, 4 | rbsyl 1764 | . . . . 5 ⊢ (¬ ¬ 𝜑 ∨ ¬ 𝜑) |
6 | rb-ax2 1761 | . . . . 5 ⊢ (¬ (¬ ¬ 𝜑 ∨ ¬ 𝜑) ∨ (¬ 𝜑 ∨ ¬ ¬ 𝜑)) | |
7 | 5, 6 | anmp 1759 | . . . 4 ⊢ (¬ 𝜑 ∨ ¬ ¬ 𝜑) |
8 | rblem2 1766 | . . . 4 ⊢ (¬ (¬ 𝜑 ∨ ¬ ¬ 𝜑) ∨ (¬ 𝜑 ∨ (¬ ¬ 𝜑 ∨ 𝜓))) | |
9 | 7, 8 | anmp 1759 | . . 3 ⊢ (¬ 𝜑 ∨ (¬ ¬ 𝜑 ∨ 𝜓)) |
10 | 2, 9 | rbsyl 1764 | . 2 ⊢ (¬ 𝜑 ∨ (¬ 𝜑 → 𝜓)) |
11 | rb-imdf 1758 | . . 3 ⊢ ¬ (¬ (¬ (𝜑 → (¬ 𝜑 → 𝜓)) ∨ (¬ 𝜑 ∨ (¬ 𝜑 → 𝜓))) ∨ ¬ (¬ (¬ 𝜑 ∨ (¬ 𝜑 → 𝜓)) ∨ (𝜑 → (¬ 𝜑 → 𝜓)))) | |
12 | 11 | rblem7 1771 | . 2 ⊢ (¬ (¬ 𝜑 ∨ (¬ 𝜑 → 𝜓)) ∨ (𝜑 → (¬ 𝜑 → 𝜓))) |
13 | 10, 12 | anmp 1759 | 1 ⊢ (𝜑 → (¬ 𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 847 |
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-or 848 |
This theorem is referenced by: (None) |
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