Mathbox for Jarvin Udandy |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > plcofph | Structured version Visualization version GIF version |
Description: Given, a,b and a "definition" for c, c is demonstrated. (Contributed by Jarvin Udandy, 8-Sep-2020.) |
Ref | Expression |
---|---|
plcofph.1 | ⊢ (𝜒 ↔ ((((𝜑 ∧ 𝜓) ↔ 𝜑) → (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))) ∧ (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑)))) |
plcofph.2 | ⊢ 𝜑 |
plcofph.3 | ⊢ 𝜓 |
Ref | Expression |
---|---|
plcofph | ⊢ 𝜒 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | plcofph.2 | . . . . 5 ⊢ 𝜑 | |
2 | pm3.24 406 | . . . . 5 ⊢ ¬ (𝜑 ∧ ¬ 𝜑) | |
3 | 1, 2 | pm3.2i 474 | . . . 4 ⊢ (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑)) |
4 | 3 | a1i 11 | . . 3 ⊢ (((𝜑 ∧ 𝜓) ↔ 𝜑) → (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))) |
5 | 4, 3 | pm3.2i 474 | . 2 ⊢ ((((𝜑 ∧ 𝜓) ↔ 𝜑) → (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))) ∧ (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))) |
6 | plcofph.1 | . . . 4 ⊢ (𝜒 ↔ ((((𝜑 ∧ 𝜓) ↔ 𝜑) → (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))) ∧ (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑)))) | |
7 | 6 | bicomi 227 | . . 3 ⊢ (((((𝜑 ∧ 𝜓) ↔ 𝜑) → (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))) ∧ (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))) ↔ 𝜒) |
8 | 7 | biimpi 219 | . 2 ⊢ (((((𝜑 ∧ 𝜓) ↔ 𝜑) → (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))) ∧ (𝜑 ∧ ¬ (𝜑 ∧ ¬ 𝜑))) → 𝜒) |
9 | 5, 8 | ax-mp 5 | 1 ⊢ 𝜒 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 |
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 |
This theorem is referenced by: plvcofph 44113 plvcofphax 44114 |
Copyright terms: Public domain | W3C validator |