Theorem confun2

Description: Confun simplified to two propositions. (Contributed by Jarvin Udandy, 6-Sep-2020)

	Ref	Expression
Hypotheses	confun2.1	⊢ ( 𝜓 → 𝜑 )
	confun2.2	⊢ ( 𝜓 → ¬ ( 𝜓 → ( 𝜓 ∧ ¬ 𝜓 ) ) )
	confun2.3	⊢ ( ( 𝜓 → 𝜑 ) → ( ( 𝜓 → 𝜑 ) → 𝜑 ) )
Assertion	confun2	⊢ ( 𝜓 → ( ¬ ( 𝜓 → ( 𝜓 ∧ ¬ 𝜓 ) ) ↔ ( 𝜓 → 𝜑 ) ) )

Proof

Step	Hyp	Ref	Expression
1		confun2.1	⊢ ( 𝜓 → 𝜑 )
2		confun2.2	⊢ ( 𝜓 → ¬ ( 𝜓 → ( 𝜓 ∧ ¬ 𝜓 ) ) )
3		confun2.3	⊢ ( ( 𝜓 → 𝜑 ) → ( ( 𝜓 → 𝜑 ) → 𝜑 ) )
4	1 1 2 3	confun	⊢ ( 𝜓 → ( ¬ ( 𝜓 → ( 𝜓 ∧ ¬ 𝜓 ) ) ↔ ( 𝜓 → 𝜑 ) ) )