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Mirrors > Home > MPE Home > Th. List > Mathboxes > rp-fakeoranass | Structured version Visualization version GIF version |
Description: A special case where a mixture of or and and appears to conform to a mixed associative law. (Contributed by RP, 29-Feb-2020.) |
Ref | Expression |
---|---|
rp-fakeoranass | ⊢ ((𝜑 → 𝜒) ↔ (((𝜑 ∨ 𝜓) ∧ 𝜒) ↔ (𝜑 ∨ (𝜓 ∧ 𝜒)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rp-fakeanorass 41792 | . 2 ⊢ ((𝜑 → 𝜒) ↔ (((𝜒 ∧ 𝜓) ∨ 𝜑) ↔ (𝜒 ∧ (𝜓 ∨ 𝜑)))) | |
2 | bicom 221 | . . 3 ⊢ ((((𝜒 ∧ 𝜓) ∨ 𝜑) ↔ (𝜒 ∧ (𝜓 ∨ 𝜑))) ↔ ((𝜒 ∧ (𝜓 ∨ 𝜑)) ↔ ((𝜒 ∧ 𝜓) ∨ 𝜑))) | |
3 | orcom 869 | . . . . 5 ⊢ ((𝜓 ∨ 𝜑) ↔ (𝜑 ∨ 𝜓)) | |
4 | 3 | anbi1ci 627 | . . . 4 ⊢ ((𝜒 ∧ (𝜓 ∨ 𝜑)) ↔ ((𝜑 ∨ 𝜓) ∧ 𝜒)) |
5 | orcom 869 | . . . . 5 ⊢ (((𝜒 ∧ 𝜓) ∨ 𝜑) ↔ (𝜑 ∨ (𝜒 ∧ 𝜓))) | |
6 | ancom 462 | . . . . . 6 ⊢ ((𝜒 ∧ 𝜓) ↔ (𝜓 ∧ 𝜒)) | |
7 | 6 | orbi2i 912 | . . . . 5 ⊢ ((𝜑 ∨ (𝜒 ∧ 𝜓)) ↔ (𝜑 ∨ (𝜓 ∧ 𝜒))) |
8 | 5, 7 | bitri 275 | . . . 4 ⊢ (((𝜒 ∧ 𝜓) ∨ 𝜑) ↔ (𝜑 ∨ (𝜓 ∧ 𝜒))) |
9 | 4, 8 | bibi12i 340 | . . 3 ⊢ (((𝜒 ∧ (𝜓 ∨ 𝜑)) ↔ ((𝜒 ∧ 𝜓) ∨ 𝜑)) ↔ (((𝜑 ∨ 𝜓) ∧ 𝜒) ↔ (𝜑 ∨ (𝜓 ∧ 𝜒)))) |
10 | 2, 9 | bitri 275 | . 2 ⊢ ((((𝜒 ∧ 𝜓) ∨ 𝜑) ↔ (𝜒 ∧ (𝜓 ∨ 𝜑))) ↔ (((𝜑 ∨ 𝜓) ∧ 𝜒) ↔ (𝜑 ∨ (𝜓 ∧ 𝜒)))) |
11 | 1, 10 | bitri 275 | 1 ⊢ ((𝜑 → 𝜒) ↔ (((𝜑 ∨ 𝜓) ∧ 𝜒) ↔ (𝜑 ∨ (𝜓 ∧ 𝜒)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 |
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 206 df-an 398 df-or 847 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |