Theorem 3anorOLD 1131

Description: Obsolete version of 3anor 1134 as of 8-Apr-2022. (Contributed by Jeff Hankins, 15-Aug-2009.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
3anorOLD	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒))

Proof of Theorem 3anorOLD

Step	Hyp	Ref	Expression
1		df-3an 1110	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒))
2		anor 1006	. . . 4 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ¬ (¬ (𝜑 ∧ 𝜓) ∨ ¬ 𝜒))
3		ianor 1005	. . . . 5 ⊢ (¬ (𝜑 ∧ 𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓))
4	3	orbi1i 938	. . . 4 ⊢ ((¬ (𝜑 ∧ 𝜓) ∨ ¬ 𝜒) ↔ ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ 𝜒))
5	2, 4	xchbinx 326	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ¬ ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ 𝜒))
6		df-3or 1109	. . 3 ⊢ ((¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒) ↔ ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ 𝜒))
7	5, 6	xchbinxr 327	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒))
8	1, 7	bitri 267	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒))

Syntax hints:  ¬ wn 3   ↔ wb 198   ∧ wa 385   ∨ wo 874   ∨ w3o 1107   ∧ w3a 1108

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 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110

This theorem is referenced by: (None)