Theorem bj-andnotim 36589

Description: Two ways of expressing a certain ternary connective. Note the respective positions of the three formulas on each side of the biconditional. (Contributed by BJ, 6-Oct-2018.)

Assertion

Ref	Expression
bj-andnotim	⊢ (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ ((𝜑 → 𝜓) ∨ 𝜒))

Proof of Theorem bj-andnotim

Step	Hyp	Ref	Expression
1		imor 854	. . 3 ⊢ (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (¬ (𝜑 ∧ ¬ 𝜓) ∨ 𝜒))
2		iman 401	. . . . 5 ⊢ ((𝜑 → 𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓))
3	2	biimpri 228	. . . 4 ⊢ (¬ (𝜑 ∧ ¬ 𝜓) → (𝜑 → 𝜓))
4	3	orim1i 910	. . 3 ⊢ ((¬ (𝜑 ∧ ¬ 𝜓) ∨ 𝜒) → ((𝜑 → 𝜓) ∨ 𝜒))
5	1, 4	sylbi 217	. 2 ⊢ (((𝜑 ∧ ¬ 𝜓) → 𝜒) → ((𝜑 → 𝜓) ∨ 𝜒))
6		pm2.24 124	. . . . 5 ⊢ (𝜓 → (¬ 𝜓 → 𝜒))
7	6	imim2i 16	. . . 4 ⊢ ((𝜑 → 𝜓) → (𝜑 → (¬ 𝜓 → 𝜒)))
8	7	impd 410	. . 3 ⊢ ((𝜑 → 𝜓) → ((𝜑 ∧ ¬ 𝜓) → 𝜒))
9		ax-1 6	. . 3 ⊢ (𝜒 → ((𝜑 ∧ ¬ 𝜓) → 𝜒))
10	8, 9	jaoi 858	. 2 ⊢ (((𝜑 → 𝜓) ∨ 𝜒) → ((𝜑 ∧ ¬ 𝜓) → 𝜒))
11	5, 10	impbii 209	1 ⊢ (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ ((𝜑 → 𝜓) ∨ 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848

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 207  df-an 396  df-or 849

This theorem is referenced by: (None)