Theorem iuneq2df

Description: Equality deduction for indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020)

	Ref	Expression
Hypotheses	iuneq2df.1	$$⊢Ⅎx\phi$$
	iuneq2df.2	$$⊢\left(\phi \wedge x\in A\right)\to B=C$$
Assertion	iuneq2df	$$⊢\phi \to \bigcup _{x\in A}B=\bigcup _{x\in A}C$$

Proof

Step	Hyp	Ref	Expression
1		iuneq2df.1	$$⊢Ⅎx\phi$$
2		iuneq2df.2	$$⊢\left(\phi \wedge x\in A\right)\to B=C$$
3	2	ex	$$⊢\phi \to \left(x\in A\to B=C\right)$$
4	1 3	ralrimi	$$⊢\phi \to \forall x\in AB=C$$
5		iuneq2	$$⊢\forall x\in AB=C\to \bigcup _{x\in A}B=\bigcup _{x\in A}C$$
6	4 5	syl	$$⊢\phi \to \bigcup _{x\in A}B=\bigcup _{x\in A}C$$