Theorem bj-andnotim

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)

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

Proof

Step	Hyp	Ref	Expression
1		imor	⊢ ( ( ( 𝜑 ∧ ¬ 𝜓 ) → 𝜒 ) ↔ ( ¬ ( 𝜑 ∧ ¬ 𝜓 ) ∨ 𝜒 ) )
2		iman	⊢ ( ( 𝜑 → 𝜓 ) ↔ ¬ ( 𝜑 ∧ ¬ 𝜓 ) )
3	2	biimpri	⊢ ( ¬ ( 𝜑 ∧ ¬ 𝜓 ) → ( 𝜑 → 𝜓 ) )
4	3	orim1i	⊢ ( ( ¬ ( 𝜑 ∧ ¬ 𝜓 ) ∨ 𝜒 ) → ( ( 𝜑 → 𝜓 ) ∨ 𝜒 ) )
5	1 4	sylbi	⊢ ( ( ( 𝜑 ∧ ¬ 𝜓 ) → 𝜒 ) → ( ( 𝜑 → 𝜓 ) ∨ 𝜒 ) )
6		pm2.24	⊢ ( 𝜓 → ( ¬ 𝜓 → 𝜒 ) )
7	6	imim2i	⊢ ( ( 𝜑 → 𝜓 ) → ( 𝜑 → ( ¬ 𝜓 → 𝜒 ) ) )
8	7	impd	⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜑 ∧ ¬ 𝜓 ) → 𝜒 ) )
9		ax-1	⊢ ( 𝜒 → ( ( 𝜑 ∧ ¬ 𝜓 ) → 𝜒 ) )
10	8 9	jaoi	⊢ ( ( ( 𝜑 → 𝜓 ) ∨ 𝜒 ) → ( ( 𝜑 ∧ ¬ 𝜓 ) → 𝜒 ) )
11	5 10	impbii	⊢ ( ( ( 𝜑 ∧ ¬ 𝜓 ) → 𝜒 ) ↔ ( ( 𝜑 → 𝜓 ) ∨ 𝜒 ) )