Theorem ifexg

Description: Conditional operator existence. (Contributed by NM, 21-Mar-2011) (Proof shortened by BJ, 1-Sep-2022)

	Ref	Expression
Assertion	ifexg	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → if ( 𝜑 , 𝐴 , 𝐵 ) ∈ V )

Proof

Step	Hyp	Ref	Expression
1		elex	⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ∈ V )
2		elex	⊢ ( 𝐵 ∈ 𝑊 → 𝐵 ∈ V )
3		ifcl	⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → if ( 𝜑 , 𝐴 , 𝐵 ) ∈ V )
4	1 2 3	syl2an	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → if ( 𝜑 , 𝐴 , 𝐵 ) ∈ V )