Theorem ifclda

Description: Conditional closure. (Contributed by Jeff Madsen, 2-Sep-2009)

	Ref	Expression
Hypotheses	ifclda.1	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝐴 ∈ 𝐶 )
	ifclda.2	⊢ ( ( 𝜑 ∧ ¬ 𝜓 ) → 𝐵 ∈ 𝐶 )
Assertion	ifclda	⊢ ( 𝜑 → if ( 𝜓 , 𝐴 , 𝐵 ) ∈ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		ifclda.1	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝐴 ∈ 𝐶 )
2		ifclda.2	⊢ ( ( 𝜑 ∧ ¬ 𝜓 ) → 𝐵 ∈ 𝐶 )
3		iftrue	⊢ ( 𝜓 → if ( 𝜓 , 𝐴 , 𝐵 ) = 𝐴 )
4	3	adantl	⊢ ( ( 𝜑 ∧ 𝜓 ) → if ( 𝜓 , 𝐴 , 𝐵 ) = 𝐴 )
5	4 1	eqeltrd	⊢ ( ( 𝜑 ∧ 𝜓 ) → if ( 𝜓 , 𝐴 , 𝐵 ) ∈ 𝐶 )
6		iffalse	⊢ ( ¬ 𝜓 → if ( 𝜓 , 𝐴 , 𝐵 ) = 𝐵 )
7	6	adantl	⊢ ( ( 𝜑 ∧ ¬ 𝜓 ) → if ( 𝜓 , 𝐴 , 𝐵 ) = 𝐵 )
8	7 2	eqeltrd	⊢ ( ( 𝜑 ∧ ¬ 𝜓 ) → if ( 𝜓 , 𝐴 , 𝐵 ) ∈ 𝐶 )
9	5 8	pm2.61dan	⊢ ( 𝜑 → if ( 𝜓 , 𝐴 , 𝐵 ) ∈ 𝐶 )