Theorem reuxfr1

Description: Transfer existential uniqueness from a variable x to another variable y contained in expression A . Use reuhyp to eliminate the second hypothesis. (Contributed by NM, 14-Nov-2004)

	Ref	Expression
Hypotheses	reuxfr1.1	|- ( y e. C -> A e. B )
	reuxfr1.2	|- ( x e. B -> E! y e. C x = A )
	reuxfr1.3	|- ( x = A -> ( ph <-> ps ) )
Assertion	reuxfr1	|- ( E! x e. B ph <-> E! y e. C ps )

Proof

Step	Hyp	Ref	Expression
1		reuxfr1.1	|- ( y e. C -> A e. B )
2		reuxfr1.2	|- ( x e. B -> E! y e. C x = A )
3		reuxfr1.3	|- ( x = A -> ( ph <-> ps ) )
4	1	adantl	|- ( ( T. /\ y e. C ) -> A e. B )
5	2	adantl	|- ( ( T. /\ x e. B ) -> E! y e. C x = A )
6	4 5 3	reuxfr1ds	|- ( T. -> ( E! x e. B ph <-> E! y e. C ps ) )
7	6	mptru	|- ( E! x e. B ph <-> E! y e. C ps )