Mathbox for Jarvin Udandy |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > confun4 | Structured version Visualization version GIF version |
Description: An attempt at derivative. Resisted simplest path to a proof. (Contributed by Jarvin Udandy, 6-Sep-2020.) |
Ref | Expression |
---|---|
confun4.1 | ⊢ 𝜑 |
confun4.2 | ⊢ ((𝜑 → 𝜓) → 𝜓) |
confun4.3 | ⊢ (𝜓 → (𝜑 → 𝜒)) |
confun4.4 | ⊢ ((𝜒 → 𝜃) → ((𝜑 → 𝜃) ↔ 𝜓)) |
confun4.5 | ⊢ (𝜏 ↔ (𝜒 → 𝜃)) |
confun4.6 | ⊢ (𝜂 ↔ ¬ (𝜒 → (𝜒 ∧ ¬ 𝜒))) |
confun4.7 | ⊢ 𝜓 |
confun4.8 | ⊢ (𝜒 → 𝜃) |
Ref | Expression |
---|---|
confun4 | ⊢ (𝜒 → (𝜓 → 𝜏)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | confun4.1 | . . . 4 ⊢ 𝜑 | |
2 | confun4.7 | . . . . 5 ⊢ 𝜓 | |
3 | confun4.3 | . . . . 5 ⊢ (𝜓 → (𝜑 → 𝜒)) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ (𝜑 → 𝜒) |
5 | 1, 4 | ax-mp 5 | . . 3 ⊢ 𝜒 |
6 | confun4.8 | . . . . . 6 ⊢ (𝜒 → 𝜃) | |
7 | confun4.5 | . . . . . . . 8 ⊢ (𝜏 ↔ (𝜒 → 𝜃)) | |
8 | bicom1 210 | . . . . . . . 8 ⊢ ((𝜏 ↔ (𝜒 → 𝜃)) → ((𝜒 → 𝜃) ↔ 𝜏)) | |
9 | 7, 8 | ax-mp 5 | . . . . . . 7 ⊢ ((𝜒 → 𝜃) ↔ 𝜏) |
10 | 9 | biimpi 205 | . . . . . 6 ⊢ ((𝜒 → 𝜃) → 𝜏) |
11 | 6, 10 | ax-mp 5 | . . . . 5 ⊢ 𝜏 |
12 | 2, 11 | pm3.2i 470 | . . . 4 ⊢ (𝜓 ∧ 𝜏) |
13 | pm3.4 582 | . . . 4 ⊢ ((𝜓 ∧ 𝜏) → (𝜓 → 𝜏)) | |
14 | 12, 13 | ax-mp 5 | . . 3 ⊢ (𝜓 → 𝜏) |
15 | 5, 14 | pm3.2i 470 | . 2 ⊢ (𝜒 ∧ (𝜓 → 𝜏)) |
16 | pm3.4 582 | . 2 ⊢ ((𝜒 ∧ (𝜓 → 𝜏)) → (𝜒 → (𝜓 → 𝜏))) | |
17 | 15, 16 | ax-mp 5 | 1 ⊢ (𝜒 → (𝜓 → 𝜏)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 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 196 df-an 385 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |