Theorem uneq12d

Description: Equality deduction for the union of two classes. (Contributed by NM, 29-Sep-2004) (Proof shortened by Andrew Salmon, 26-Jun-2011)

	Ref	Expression
Hypotheses	uneq1d.1	$$⊢\phi \to A=B$$
	uneq12d.2	$$⊢\phi \to C=D$$
Assertion	uneq12d	$$⊢\phi \to A\cup C=B\cup D$$

Proof

Step	Hyp	Ref	Expression
1		uneq1d.1	$$⊢\phi \to A=B$$
2		uneq12d.2	$$⊢\phi \to C=D$$
3		uneq12	$$⊢\left(A=B\wedge C=D\right)\to A\cup C=B\cup D$$
4	1 2 3	syl2anc	$$⊢\phi \to A\cup C=B\cup D$$