Theorem reusv3i

Description: Two ways of expressing existential uniqueness via an indirect equality. (Contributed by NM, 23-Dec-2012)

	Ref	Expression
Hypotheses	reusv3.1	|- ( y = z -> ( ph <-> ps ) )
	reusv3.2	|- ( y = z -> C = D )
Assertion	reusv3i	|- ( E. x e. A A. y e. B ( ph -> x = C ) -> A. y e. B A. z e. B ( ( ph /\ ps ) -> C = D ) )

Proof

Step	Hyp	Ref	Expression
1		reusv3.1	|- ( y = z -> ( ph <-> ps ) )
2		reusv3.2	|- ( y = z -> C = D )
3	2	eqeq2d	|- ( y = z -> ( x = C <-> x = D ) )
4	1 3	imbi12d	|- ( y = z -> ( ( ph -> x = C ) <-> ( ps -> x = D ) ) )
5	4	cbvralvw	|- ( A. y e. B ( ph -> x = C ) <-> A. z e. B ( ps -> x = D ) )
6	5	biimpi	|- ( A. y e. B ( ph -> x = C ) -> A. z e. B ( ps -> x = D ) )
7		raaanv	|- ( A. y e. B A. z e. B ( ( ph -> x = C ) /\ ( ps -> x = D ) ) <-> ( A. y e. B ( ph -> x = C ) /\ A. z e. B ( ps -> x = D ) ) )
8		anim12	|- ( ( ( ph -> x = C ) /\ ( ps -> x = D ) ) -> ( ( ph /\ ps ) -> ( x = C /\ x = D ) ) )
9		eqtr2	|- ( ( x = C /\ x = D ) -> C = D )
10	8 9	syl6	|- ( ( ( ph -> x = C ) /\ ( ps -> x = D ) ) -> ( ( ph /\ ps ) -> C = D ) )
11	10	2ralimi	|- ( A. y e. B A. z e. B ( ( ph -> x = C ) /\ ( ps -> x = D ) ) -> A. y e. B A. z e. B ( ( ph /\ ps ) -> C = D ) )
12	7 11	sylbir	|- ( ( A. y e. B ( ph -> x = C ) /\ A. z e. B ( ps -> x = D ) ) -> A. y e. B A. z e. B ( ( ph /\ ps ) -> C = D ) )
13	6 12	mpdan	|- ( A. y e. B ( ph -> x = C ) -> A. y e. B A. z e. B ( ( ph /\ ps ) -> C = D ) )
14	13	rexlimivw	|- ( E. x e. A A. y e. B ( ph -> x = C ) -> A. y e. B A. z e. B ( ( ph /\ ps ) -> C = D ) )