Theorem noranOLD 1527

Description: Obsolete version of noran 1526 as of 8-Dec-2023. (Contributed by Remi, 26-Oct-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
noranOLD	⊢ ((𝜑 ∧ 𝜓) ↔ ((𝜑 ⊽ 𝜑) ⊽ (𝜓 ⊽ 𝜓)))

Proof of Theorem noranOLD

Step	Hyp	Ref	Expression
1		ianor 978	. . . 4 ⊢ (¬ (𝜑 ∧ 𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓))
2	1	notbii 322	. . 3 ⊢ (¬ ¬ (𝜑 ∧ 𝜓) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓))
3		nornot 1524	. . . . 5 ⊢ (¬ 𝜑 ↔ (𝜑 ⊽ 𝜑))
4		nornot 1524	. . . . 5 ⊢ (¬ 𝜓 ↔ (𝜓 ⊽ 𝜓))
5	3, 4	orbi12i 911	. . . 4 ⊢ ((¬ 𝜑 ∨ ¬ 𝜓) ↔ ((𝜑 ⊽ 𝜑) ∨ (𝜓 ⊽ 𝜓)))
6	5	notbii 322	. . 3 ⊢ (¬ (¬ 𝜑 ∨ ¬ 𝜓) ↔ ¬ ((𝜑 ⊽ 𝜑) ∨ (𝜓 ⊽ 𝜓)))
7		ioran 980	. . 3 ⊢ (¬ ((𝜑 ⊽ 𝜑) ∨ (𝜓 ⊽ 𝜓)) ↔ (¬ (𝜑 ⊽ 𝜑) ∧ ¬ (𝜓 ⊽ 𝜓)))
8	2, 6, 7	3bitrri 300	. 2 ⊢ ((¬ (𝜑 ⊽ 𝜑) ∧ ¬ (𝜓 ⊽ 𝜓)) ↔ ¬ ¬ (𝜑 ∧ 𝜓))
9		df-nor 1521	. . 3 ⊢ (((𝜑 ⊽ 𝜑) ⊽ (𝜓 ⊽ 𝜓)) ↔ ¬ ((𝜑 ⊽ 𝜑) ∨ (𝜓 ⊽ 𝜓)))
10	9, 7	bitri 277	. 2 ⊢ (((𝜑 ⊽ 𝜑) ⊽ (𝜓 ⊽ 𝜓)) ↔ (¬ (𝜑 ⊽ 𝜑) ∧ ¬ (𝜓 ⊽ 𝜓)))
11		notnotb 317	. 2 ⊢ ((𝜑 ∧ 𝜓) ↔ ¬ ¬ (𝜑 ∧ 𝜓))
12	8, 10, 11	3bitr4ri 306	1 ⊢ ((𝜑 ∧ 𝜓) ↔ ((𝜑 ⊽ 𝜑) ⊽ (𝜓 ⊽ 𝜓)))

Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 398   ∨ wo 843   ⊽ wnor 1520

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 209  df-an 399  df-or 844  df-nor 1521

This theorem is referenced by: (None)