Metamath Proof Explorer

Theorem norass

Description: A characterization of when an expression involving joint denials associates. This is identical to the case when alternative denial is associative, see nanass . Remark: Like alternative denial, joint denial is also commutative, see norcom . (Contributed by RP, 29-Oct-2023) (Proof shortened by Wolf Lammen, 17-Dec-2023)

Ref Expression
Assertion norass ( ( 𝜑 ↔ 𝜒 ) ↔ ( ( ( 𝜑 ⊽ 𝜓 ) ⊽ 𝜒 ) ↔ ( 𝜑 ⊽ ( 𝜓 ⊽ 𝜒 ) ) ) )

Proof

Step Hyp Ref Expression
1 notbi ⊢ ( ( ( ( 𝜑 ⊽ 𝜓 ) ∨ 𝜒 ) ↔ ( 𝜑 ∨ ( 𝜓 ⊽ 𝜒 ) ) ) ↔ ( ¬ ( ( 𝜑 ⊽ 𝜓 ) ∨ 𝜒 ) ↔ ¬ ( 𝜑 ∨ ( 𝜓 ⊽ 𝜒 ) ) ) )
2 norasslem1 ⊢ ( ( ( 𝜓 ∨ 𝜑 ) → 𝜒 ) ↔ ( ( 𝜓 ⊽ 𝜑 ) ∨ 𝜒 ) )
3 norasslem1 ⊢ ( ( ( 𝜓 ∨ 𝜒 ) → 𝜑 ) ↔ ( ( 𝜓 ⊽ 𝜒 ) ∨ 𝜑 ) )
4 2 3 bibi12i ⊢ ( ( ( ( 𝜓 ∨ 𝜑 ) → 𝜒 ) ↔ ( ( 𝜓 ∨ 𝜒 ) → 𝜑 ) ) ↔ ( ( ( 𝜓 ⊽ 𝜑 ) ∨ 𝜒 ) ↔ ( ( 𝜓 ⊽ 𝜒 ) ∨ 𝜑 ) ) )
5 bicom ⊢ ( ( 𝜑 ↔ 𝜒 ) ↔ ( 𝜒 ↔ 𝜑 ) )
6 norasslem2 ⊢ ( 𝜓 → ( 𝜒 ↔ ( ( 𝜓 ∨ 𝜑 ) → 𝜒 ) ) )
7 norasslem2 ⊢ ( 𝜓 → ( 𝜑 ↔ ( ( 𝜓 ∨ 𝜒 ) → 𝜑 ) ) )
8 6 7 bibi12d ⊢ ( 𝜓 → ( ( 𝜒 ↔ 𝜑 ) ↔ ( ( ( 𝜓 ∨ 𝜑 ) → 𝜒 ) ↔ ( ( 𝜓 ∨ 𝜒 ) → 𝜑 ) ) ) )
9 5 8 bitrid ⊢ ( 𝜓 → ( ( 𝜑 ↔ 𝜒 ) ↔ ( ( ( 𝜓 ∨ 𝜑 ) → 𝜒 ) ↔ ( ( 𝜓 ∨ 𝜒 ) → 𝜑 ) ) ) )
10 impimprbi ⊢ ( ( 𝜑 ↔ 𝜒 ) ↔ ( ( 𝜑 → 𝜒 ) ↔ ( 𝜒 → 𝜑 ) ) )
11 norasslem3 ⊢ ( ¬ 𝜓 → ( ( 𝜑 → 𝜒 ) ↔ ( ( 𝜓 ∨ 𝜑 ) → 𝜒 ) ) )
12 norasslem3 ⊢ ( ¬ 𝜓 → ( ( 𝜒 → 𝜑 ) ↔ ( ( 𝜓 ∨ 𝜒 ) → 𝜑 ) ) )
13 11 12 bibi12d ⊢ ( ¬ 𝜓 → ( ( ( 𝜑 → 𝜒 ) ↔ ( 𝜒 → 𝜑 ) ) ↔ ( ( ( 𝜓 ∨ 𝜑 ) → 𝜒 ) ↔ ( ( 𝜓 ∨ 𝜒 ) → 𝜑 ) ) ) )
14 10 13 bitrid ⊢ ( ¬ 𝜓 → ( ( 𝜑 ↔ 𝜒 ) ↔ ( ( ( 𝜓 ∨ 𝜑 ) → 𝜒 ) ↔ ( ( 𝜓 ∨ 𝜒 ) → 𝜑 ) ) ) )
15 9 14 pm2.61i ⊢ ( ( 𝜑 ↔ 𝜒 ) ↔ ( ( ( 𝜓 ∨ 𝜑 ) → 𝜒 ) ↔ ( ( 𝜓 ∨ 𝜒 ) → 𝜑 ) ) )
16 norcom ⊢ ( ( 𝜑 ⊽ 𝜓 ) ↔ ( 𝜓 ⊽ 𝜑 ) )
17 16 orbi1i ⊢ ( ( ( 𝜑 ⊽ 𝜓 ) ∨ 𝜒 ) ↔ ( ( 𝜓 ⊽ 𝜑 ) ∨ 𝜒 ) )
18 orcom ⊢ ( ( 𝜑 ∨ ( 𝜓 ⊽ 𝜒 ) ) ↔ ( ( 𝜓 ⊽ 𝜒 ) ∨ 𝜑 ) )
19 17 18 bibi12i ⊢ ( ( ( ( 𝜑 ⊽ 𝜓 ) ∨ 𝜒 ) ↔ ( 𝜑 ∨ ( 𝜓 ⊽ 𝜒 ) ) ) ↔ ( ( ( 𝜓 ⊽ 𝜑 ) ∨ 𝜒 ) ↔ ( ( 𝜓 ⊽ 𝜒 ) ∨ 𝜑 ) ) )
20 4 15 19 3bitr4i ⊢ ( ( 𝜑 ↔ 𝜒 ) ↔ ( ( ( 𝜑 ⊽ 𝜓 ) ∨ 𝜒 ) ↔ ( 𝜑 ∨ ( 𝜓 ⊽ 𝜒 ) ) ) )
21 df-nor ⊢ ( ( ( 𝜑 ⊽ 𝜓 ) ⊽ 𝜒 ) ↔ ¬ ( ( 𝜑 ⊽ 𝜓 ) ∨ 𝜒 ) )
22 df-nor ⊢ ( ( 𝜑 ⊽ ( 𝜓 ⊽ 𝜒 ) ) ↔ ¬ ( 𝜑 ∨ ( 𝜓 ⊽ 𝜒 ) ) )
23 21 22 bibi12i ⊢ ( ( ( ( 𝜑 ⊽ 𝜓 ) ⊽ 𝜒 ) ↔ ( 𝜑 ⊽ ( 𝜓 ⊽ 𝜒 ) ) ) ↔ ( ¬ ( ( 𝜑 ⊽ 𝜓 ) ∨ 𝜒 ) ↔ ¬ ( 𝜑 ∨ ( 𝜓 ⊽ 𝜒 ) ) ) )
24 1 20 23 3bitr4i ⊢ ( ( 𝜑 ↔ 𝜒 ) ↔ ( ( ( 𝜑 ⊽ 𝜓 ) ⊽ 𝜒 ) ↔ ( 𝜑 ⊽ ( 𝜓 ⊽ 𝜒 ) ) ) )