Theorem nfreuw

Description: Bound-variable hypothesis builder for restricted unique existence. Version of nfreu with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 30-Oct-2010) (Revised by Gino Giotto, 10-Jan-2024) Avoid ax-9 , ax-ext . (Revised by Wolf Lammen, 21-Nov-2024)

	Ref	Expression
Hypotheses	nfreuw.1	|- F/_ x A
	nfreuw.2	|- F/ x ph
Assertion	nfreuw	|- F/ x E! y e. A ph

Proof

Step	Hyp	Ref	Expression
1		nfreuw.1	|- F/_ x A
2		nfreuw.2	|- F/ x ph
3		df-reu	|- ( E! y e. A ph <-> E! y ( y e. A /\ ph ) )
4	1	nfcri	|- F/ x y e. A
5	4 2	nfan	|- F/ x ( y e. A /\ ph )
6	5	nfeuw	|- F/ x E! y ( y e. A /\ ph )
7	3 6	nfxfr	|- F/ x E! y e. A ph