Theorem confun3

Description: Confun's more complex form where both a,d have been "defined". (Contributed by Jarvin Udandy, 6-Sep-2020)

	Ref	Expression
Hypotheses	confun3.1	⊢ ( 𝜑 ↔ ( 𝜒 → 𝜓 ) )
	confun3.2	⊢ ( 𝜃 ↔ ¬ ( 𝜒 → ( 𝜒 ∧ ¬ 𝜒 ) ) )
	confun3.3	⊢ ( 𝜒 → 𝜓 )
	confun3.4	⊢ ( 𝜒 → ¬ ( 𝜒 → ( 𝜒 ∧ ¬ 𝜒 ) ) )
	confun3.5	⊢ ( ( 𝜒 → 𝜓 ) → ( ( 𝜒 → 𝜓 ) → 𝜓 ) )
Assertion	confun3	⊢ ( 𝜒 → ( ¬ ( 𝜒 → ( 𝜒 ∧ ¬ 𝜒 ) ) ↔ ( 𝜒 → 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		confun3.1	⊢ ( 𝜑 ↔ ( 𝜒 → 𝜓 ) )
2		confun3.2	⊢ ( 𝜃 ↔ ¬ ( 𝜒 → ( 𝜒 ∧ ¬ 𝜒 ) ) )
3		confun3.3	⊢ ( 𝜒 → 𝜓 )
4		confun3.4	⊢ ( 𝜒 → ¬ ( 𝜒 → ( 𝜒 ∧ ¬ 𝜒 ) ) )
5		confun3.5	⊢ ( ( 𝜒 → 𝜓 ) → ( ( 𝜒 → 𝜓 ) → 𝜓 ) )
6	3 3 4 5	confun	⊢ ( 𝜒 → ( ¬ ( 𝜒 → ( 𝜒 ∧ ¬ 𝜒 ) ) ↔ ( 𝜒 → 𝜓 ) ) )