Theorem eqif

Description: Expansion of an equality with a conditional operator. (Contributed by NM, 14-Feb-2005)

	Ref	Expression
Assertion	eqif	⊢ ( 𝐴 = if ( 𝜑 , 𝐵 , 𝐶 ) ↔ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) ∨ ( ¬ 𝜑 ∧ 𝐴 = 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		eqeq2	⊢ ( if ( 𝜑 , 𝐵 , 𝐶 ) = 𝐵 → ( 𝐴 = if ( 𝜑 , 𝐵 , 𝐶 ) ↔ 𝐴 = 𝐵 ) )
2		eqeq2	⊢ ( if ( 𝜑 , 𝐵 , 𝐶 ) = 𝐶 → ( 𝐴 = if ( 𝜑 , 𝐵 , 𝐶 ) ↔ 𝐴 = 𝐶 ) )
3	1 2	elimif	⊢ ( 𝐴 = if ( 𝜑 , 𝐵 , 𝐶 ) ↔ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) ∨ ( ¬ 𝜑 ∧ 𝐴 = 𝐶 ) ) )