Theorem clifte

Description: show d is the same as an if-else involving a,b. (Contributed by Jarvin Udandy, 20-Sep-2020)

	Ref	Expression
Hypotheses	clifte.1	⊢ ( 𝜑 ∧ ¬ 𝜒 )
	clifte.2	⊢ 𝜃
Assertion	clifte	⊢ ( 𝜃 ↔ ( ( 𝜑 ∧ ¬ 𝜒 ) ∨ ( 𝜓 ∧ 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		clifte.1	⊢ ( 𝜑 ∧ ¬ 𝜒 )
2		clifte.2	⊢ 𝜃
3	1	orci	⊢ ( ( 𝜑 ∧ ¬ 𝜒 ) ∨ ( 𝜓 ∧ 𝜒 ) )
4	2 3	2th	⊢ ( 𝜃 ↔ ( ( 𝜑 ∧ ¬ 𝜒 ) ∨ ( 𝜓 ∧ 𝜒 ) ) )