Theorem wl-ifp-ncond1

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-ncond1	⊢ ( ¬ 𝜓 → ( if- ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( ¬ 𝜑 ∧ 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		df-ifp	⊢ ( if- ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∨ ( ¬ 𝜑 ∧ 𝜒 ) ) )
2		simpr	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜓 )
3	2	con3i	⊢ ( ¬ 𝜓 → ¬ ( 𝜑 ∧ 𝜓 ) )
4		biorf	⊢ ( ¬ ( 𝜑 ∧ 𝜓 ) → ( ( ¬ 𝜑 ∧ 𝜒 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∨ ( ¬ 𝜑 ∧ 𝜒 ) ) ) )
5	3 4	syl	⊢ ( ¬ 𝜓 → ( ( ¬ 𝜑 ∧ 𝜒 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∨ ( ¬ 𝜑 ∧ 𝜒 ) ) ) )
6	1 5	bitr4id	⊢ ( ¬ 𝜓 → ( if- ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( ¬ 𝜑 ∧ 𝜒 ) ) )