Theorem iuneq12f

Description: Equality deduction for indexed unions. (Contributed by Giovanni Mascellani, 10-Apr-2018)

	Ref	Expression
Hypotheses	iuneq12f.1	$$⊢\underset{\_}{Ⅎ}xA$$
	iuneq12f.2	$$⊢\underset{\_}{Ⅎ}xB$$
Assertion	iuneq12f	$$⊢\left(A=B\wedge \forall x\in AC=D\right)\to \bigcup _{x\in A}C=\bigcup _{x\in B}D$$

Proof

Step	Hyp	Ref	Expression
1		iuneq12f.1	$$⊢\underset{\_}{Ⅎ}xA$$
2		iuneq12f.2	$$⊢\underset{\_}{Ⅎ}xB$$
3		iuneq2	$$⊢\forall x\in AC=D\to \bigcup _{x\in A}C=\bigcup _{x\in A}D$$
4	1 2	iuneq2f	$$⊢A=B\to \bigcup _{x\in A}D=\bigcup _{x\in B}D$$
5	3 4	sylan9eqr	$$⊢\left(A=B\wedge \forall x\in AC=D\right)\to \bigcup _{x\in A}C=\bigcup _{x\in B}D$$