Theorem annotanannotOLD 1035

Description: Obsolete proof of annotanannot 864 as of 1-Apr-2022. (Contributed by AV, 8-Mar-2022.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
annotanannotOLD	⊢ ((𝜑 ∧ ¬ (𝜑 ∧ 𝜓)) ↔ (𝜑 ∧ ¬ 𝜓))

Proof of Theorem annotanannotOLD

Step	Hyp	Ref	Expression
1		ianor 1005	. . 3 ⊢ (¬ (𝜑 ∧ 𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓))
2	1	anbi2i 617	. 2 ⊢ ((𝜑 ∧ ¬ (𝜑 ∧ 𝜓)) ↔ (𝜑 ∧ (¬ 𝜑 ∨ ¬ 𝜓)))
3		andi 1031	. 2 ⊢ ((𝜑 ∧ (¬ 𝜑 ∨ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ 𝜑) ∨ (𝜑 ∧ ¬ 𝜓)))
4		pm3.24 392	. . . . 5 ⊢ ¬ (𝜑 ∧ ¬ 𝜑)
5	4	pm2.21i 117	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝜑) → (𝜑 ∧ ¬ 𝜓))
6		id 22	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝜓) → (𝜑 ∧ ¬ 𝜓))
7	5, 6	jaoi 884	. . 3 ⊢ (((𝜑 ∧ ¬ 𝜑) ∨ (𝜑 ∧ ¬ 𝜓)) → (𝜑 ∧ ¬ 𝜓))
8		olc 895	. . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → ((𝜑 ∧ ¬ 𝜑) ∨ (𝜑 ∧ ¬ 𝜓)))
9	7, 8	impbii 201	. 2 ⊢ (((𝜑 ∧ ¬ 𝜑) ∨ (𝜑 ∧ ¬ 𝜓)) ↔ (𝜑 ∧ ¬ 𝜓))
10	2, 3, 9	3bitri 289	1 ⊢ ((𝜑 ∧ ¬ (𝜑 ∧ 𝜓)) ↔ (𝜑 ∧ ¬ 𝜓))

Syntax hints:  ¬ wn 3   ↔ wb 198   ∧ wa 385   ∨ wo 874

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

This theorem is referenced by: (None)