Theorem ifexg

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

	Ref	Expression
Assertion	ifexg	$$⊢\left(A\in V\wedge B\in W\right)\to if\left(\phi ,A,B\right)\in \mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		elex	$$⊢A\in V\to A\in \mathrm{V}$$
2		elex	$$⊢B\in W\to B\in \mathrm{V}$$
3		ifcl	$$⊢\left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right)\to if\left(\phi ,A,B\right)\in \mathrm{V}$$
4	1 2 3	syl2an	$$⊢\left(A\in V\wedge B\in W\right)\to if\left(\phi ,A,B\right)\in \mathrm{V}$$