Description: A single axiom for propositional calculus discovered by C. A. Meredith.
This axiom has 19 symbols, sans auxiliaries. See notes in merco1 . (Contributed by Anthony Hart, 7-Aug-2011) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | merco2 | ⊢ ( ( ( 𝜑 → 𝜓 ) → ( ( ⊥ → 𝜒 ) → 𝜃 ) ) → ( ( 𝜃 → 𝜑 ) → ( 𝜏 → ( 𝜂 → 𝜑 ) ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | falim | ⊢ ( ⊥ → 𝜒 ) | |
| 2 | pm2.04 | ⊢ ( ( ( 𝜑 → 𝜓 ) → ( ( ⊥ → 𝜒 ) → 𝜃 ) ) → ( ( ⊥ → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → 𝜃 ) ) ) | |
| 3 | 1 2 | mpi | ⊢ ( ( ( 𝜑 → 𝜓 ) → ( ( ⊥ → 𝜒 ) → 𝜃 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜃 ) ) |
| 4 | jarl | ⊢ ( ( ( 𝜑 → 𝜓 ) → 𝜃 ) → ( ¬ 𝜑 → 𝜃 ) ) | |
| 5 | idd | ⊢ ( ( ( 𝜑 → 𝜓 ) → 𝜃 ) → ( 𝜃 → 𝜃 ) ) | |
| 6 | 4 5 | jad | ⊢ ( ( ( 𝜑 → 𝜓 ) → 𝜃 ) → ( ( 𝜑 → 𝜃 ) → 𝜃 ) ) |
| 7 | looinv | ⊢ ( ( ( 𝜑 → 𝜃 ) → 𝜃 ) → ( ( 𝜃 → 𝜑 ) → 𝜑 ) ) | |
| 8 | 3 6 7 | 3syl | ⊢ ( ( ( 𝜑 → 𝜓 ) → ( ( ⊥ → 𝜒 ) → 𝜃 ) ) → ( ( 𝜃 → 𝜑 ) → 𝜑 ) ) |
| 9 | 8 | a1dd | ⊢ ( ( ( 𝜑 → 𝜓 ) → ( ( ⊥ → 𝜒 ) → 𝜃 ) ) → ( ( 𝜃 → 𝜑 ) → ( 𝜏 → 𝜑 ) ) ) |
| 10 | 9 | a1i | ⊢ ( 𝜂 → ( ( ( 𝜑 → 𝜓 ) → ( ( ⊥ → 𝜒 ) → 𝜃 ) ) → ( ( 𝜃 → 𝜑 ) → ( 𝜏 → 𝜑 ) ) ) ) |
| 11 | 10 | com4l | ⊢ ( ( ( 𝜑 → 𝜓 ) → ( ( ⊥ → 𝜒 ) → 𝜃 ) ) → ( ( 𝜃 → 𝜑 ) → ( 𝜏 → ( 𝜂 → 𝜑 ) ) ) ) |