Theorem wl-ifp-ncond2

Description: If one case of an if- condition is false, the other automatically follows. (Contributed by Wolf Lammen, 21-Jul-2024)

	Ref	Expression
Assertion	wl-ifp-ncond2	⊢ ( ¬ 𝜒 → ( if- ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( 𝜑 ∧ 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		wl-ifp-ncond1	⊢ ( ¬ 𝜒 → ( if- ( ¬ 𝜑 , 𝜒 , 𝜓 ) ↔ ( ¬ ¬ 𝜑 ∧ 𝜓 ) ) )
2		ifpn	⊢ ( if- ( 𝜑 , 𝜓 , 𝜒 ) ↔ if- ( ¬ 𝜑 , 𝜒 , 𝜓 ) )
3		notnotb	⊢ ( 𝜑 ↔ ¬ ¬ 𝜑 )
4	3	anbi1i	⊢ ( ( 𝜑 ∧ 𝜓 ) ↔ ( ¬ ¬ 𝜑 ∧ 𝜓 ) )
5	1 2 4	3bitr4g	⊢ ( ¬ 𝜒 → ( if- ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( 𝜑 ∧ 𝜓 ) ) )