Theorem iuneq2df

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

	Ref	Expression
Hypotheses	iuneq2df.1	|- F/ x ph
	iuneq2df.2	|- ( ( ph /\ x e. A ) -> B = C )
Assertion	iuneq2df	|- ( ph -> U_ x e. A B = U_ x e. A C )

Proof

Step	Hyp	Ref	Expression
1		iuneq2df.1	|- F/ x ph
2		iuneq2df.2	|- ( ( ph /\ x e. A ) -> B = C )
3	2	ex	|- ( ph -> ( x e. A -> B = C ) )
4	1 3	ralrimi	|- ( ph -> A. x e. A B = C )
5		iuneq2	|- ( A. x e. A B = C -> U_ x e. A B = U_ x e. A C )
6	4 5	syl	|- ( ph -> U_ x e. A B = U_ x e. A C )