Theorem iuneq12d

Description: Equality deduction for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015)

	Ref	Expression
Hypotheses	iuneq1d.1	$$⊢\phi \to A=B$$
	iuneq12d.2	$$⊢\phi \to C=D$$
Assertion	iuneq12d	$$⊢\phi \to \bigcup _{x\in A}C=\bigcup _{x\in B}D$$

Proof

Step	Hyp	Ref	Expression
1		iuneq1d.1	$$⊢\phi \to A=B$$
2		iuneq12d.2	$$⊢\phi \to C=D$$
3	1	iuneq1d	$$⊢\phi \to \bigcup _{x\in A}C=\bigcup _{x\in B}C$$
4	2	adantr	$$⊢\left(\phi \wedge x\in B\right)\to C=D$$
5	4	iuneq2dv	$$⊢\phi \to \bigcup _{x\in B}C=\bigcup _{x\in B}D$$
6	3 5	eqtrd	$$⊢\phi \to \bigcup _{x\in A}C=\bigcup _{x\in B}D$$