Theorem iuneq12df

Description: Equality deduction for indexed union, deduction version. (Contributed by Thierry Arnoux, 31-Dec-2016)

	Ref	Expression
Hypotheses	iuneq12df.1	|- F/ x ph
	iuneq12df.2	|- F/_ x A
	iuneq12df.3	|- F/_ x B
	iuneq12df.4	|- ( ph -> A = B )
	iuneq12df.5	|- ( ph -> C = D )
Assertion	iuneq12df	|- ( ph -> U_ x e. A C = U_ x e. B D )

Proof

Step	Hyp	Ref	Expression
1		iuneq12df.1	|- F/ x ph
2		iuneq12df.2	|- F/_ x A
3		iuneq12df.3	|- F/_ x B
4		iuneq12df.4	|- ( ph -> A = B )
5		iuneq12df.5	|- ( ph -> C = D )
6	5	eleq2d	|- ( ph -> ( y e. C <-> y e. D ) )
7	1 2 3 4 6	rexeqbid	|- ( ph -> ( E. x e. A y e. C <-> E. x e. B y e. D ) )
8	7	alrimiv	|- ( ph -> A. y ( E. x e. A y e. C <-> E. x e. B y e. D ) )
9		abbi1	|- ( A. y ( E. x e. A y e. C <-> E. x e. B y e. D ) -> { y | E. x e. A y e. C } = { y | E. x e. B y e. D } )
10		df-iun	|- U_ x e. A C = { y | E. x e. A y e. C }
11		df-iun	|- U_ x e. B D = { y | E. x e. B y e. D }
12	9 10 11	3eqtr4g	|- ( A. y ( E. x e. A y e. C <-> E. x e. B y e. D ) -> U_ x e. A C = U_ x e. B D )
13	8 12	syl	|- ( ph -> U_ x e. A C = U_ x e. B D )