Theorem ifexg

Description: Existence of the conditional operator (closed form). (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		simpl	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → 𝐴 ∈ 𝑉 )
2		simpr	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → 𝐵 ∈ 𝑊 )
3	1 2	ifexd	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → if ( 𝜑 , 𝐴 , 𝐵 ) ∈ V )