Theorem noran 1525

Description: ∧ is expressible via ⊽. (Contributed by Remi, 26-Oct-2023.) (Proof shortened by Wolf Lammen, 8-Dec-2023.)

Assertion

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

Proof of Theorem noran

Step	Hyp	Ref	Expression
1		anor 979	. . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓))
2		nornot 1523	. . . 4 ⊢ (¬ 𝜑 ↔ (𝜑 ⊽ 𝜑))
3		nornot 1523	. . . 4 ⊢ (¬ 𝜓 ↔ (𝜓 ⊽ 𝜓))
4	2, 3	orbi12i 911	. . 3 ⊢ ((¬ 𝜑 ∨ ¬ 𝜓) ↔ ((𝜑 ⊽ 𝜑) ∨ (𝜓 ⊽ 𝜓)))
5	1, 4	xchbinx 336	. 2 ⊢ ((𝜑 ∧ 𝜓) ↔ ¬ ((𝜑 ⊽ 𝜑) ∨ (𝜓 ⊽ 𝜓)))
6		df-nor 1520	. 2 ⊢ (((𝜑 ⊽ 𝜑) ⊽ (𝜓 ⊽ 𝜓)) ↔ ¬ ((𝜑 ⊽ 𝜑) ∨ (𝜓 ⊽ 𝜓)))
7	5, 6	bitr4i 280	1 ⊢ ((𝜑 ∧ 𝜓) ↔ ((𝜑 ⊽ 𝜑) ⊽ (𝜓 ⊽ 𝜓)))

Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 398   ∨ wo 843   ⊽ wnor 1519

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 1520

This theorem is referenced by: (None)