Theorem eluni2f

Description: Membership in class union. Restricted quantifier version. (Contributed by Glauco Siliprandi, 26-Jun-2021)

	Ref	Expression
Hypotheses	eluni2f.1	$$⊢\underset{\_}{Ⅎ}xA$$
	eluni2f.2	$$⊢\underset{\_}{Ⅎ}xB$$
Assertion	eluni2f	$$⊢A\in \bigcup B\leftrightarrow \exists x\in BA\in x$$

Proof

Step	Hyp	Ref	Expression
1		eluni2f.1	$$⊢\underset{\_}{Ⅎ}xA$$
2		eluni2f.2	$$⊢\underset{\_}{Ⅎ}xB$$
3		exancom	$$⊢\exists x\left(A\in x\wedge x\in B\right)\leftrightarrow \exists x\left(x\in B\wedge A\in x\right)$$
4	1 2	elunif	$$⊢A\in \bigcup B\leftrightarrow \exists x\left(A\in x\wedge x\in B\right)$$
5		df-rex	$$⊢\exists x\in BA\in x\leftrightarrow \exists x\left(x\in B\wedge A\in x\right)$$
6	3 4 5	3bitr4i	$$⊢A\in \bigcup B\leftrightarrow \exists x\in BA\in x$$