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Mirrors > Home > MPE Home > Th. List > merco2 | Structured version Visualization version GIF version |
Description: A single axiom for
propositional calculus offered by Meredith.
This axiom has 19 symbols, sans auxiliaries. See notes in merco1 1629. (Contributed by Anthony Hart, 7-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
merco2 | ⊢ (((𝜑 → 𝜓) → ((⊥ → 𝜒) → 𝜃)) → ((𝜃 → 𝜑) → (𝜏 → (𝜂 → 𝜑)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | falim 1489 | . . . . . 6 ⊢ (⊥ → 𝜒) | |
2 | pm2.04 88 | . . . . . 6 ⊢ (((𝜑 → 𝜓) → ((⊥ → 𝜒) → 𝜃)) → ((⊥ → 𝜒) → ((𝜑 → 𝜓) → 𝜃))) | |
3 | 1, 2 | mpi 20 | . . . . 5 ⊢ (((𝜑 → 𝜓) → ((⊥ → 𝜒) → 𝜃)) → ((𝜑 → 𝜓) → 𝜃)) |
4 | jarl 174 | . . . . . 6 ⊢ (((𝜑 → 𝜓) → 𝜃) → (¬ 𝜑 → 𝜃)) | |
5 | idd 24 | . . . . . 6 ⊢ (((𝜑 → 𝜓) → 𝜃) → (𝜃 → 𝜃)) | |
6 | 4, 5 | jad 173 | . . . . 5 ⊢ (((𝜑 → 𝜓) → 𝜃) → ((𝜑 → 𝜃) → 𝜃)) |
7 | looinv 193 | . . . . 5 ⊢ (((𝜑 → 𝜃) → 𝜃) → ((𝜃 → 𝜑) → 𝜑)) | |
8 | 3, 6, 7 | 3syl 18 | . . . 4 ⊢ (((𝜑 → 𝜓) → ((⊥ → 𝜒) → 𝜃)) → ((𝜃 → 𝜑) → 𝜑)) |
9 | 8 | a1dd 48 | . . 3 ⊢ (((𝜑 → 𝜓) → ((⊥ → 𝜒) → 𝜃)) → ((𝜃 → 𝜑) → (𝜏 → 𝜑))) |
10 | 9 | a1i 11 | . 2 ⊢ (𝜂 → (((𝜑 → 𝜓) → ((⊥ → 𝜒) → 𝜃)) → ((𝜃 → 𝜑) → (𝜏 → 𝜑)))) |
11 | 10 | com4l 90 | 1 ⊢ (((𝜑 → 𝜓) → ((⊥ → 𝜒) → 𝜃)) → ((𝜃 → 𝜑) → (𝜏 → (𝜂 → 𝜑)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊥wfal 1480 |
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 196 df-tru 1478 df-fal 1481 |
This theorem is referenced by: mercolem1 1653 mercolem2 1654 mercolem3 1655 mercolem4 1656 mercolem5 1657 mercolem6 1658 mercolem7 1659 mercolem8 1660 re1tbw4 1664 |
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