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)

	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		nftru	|- F/ y T.
5		nfcvd	|- ( T. -> F/_ x y )
6	1	a1i	|- ( T. -> F/_ x A )
7	5 6	nfeld	|- ( T. -> F/ x y e. A )
8	2	a1i	|- ( T. -> F/ x ph )
9	7 8	nfand	|- ( T. -> F/ x ( y e. A /\ ph ) )
10	4 9	nfeudw	|- ( T. -> F/ x E! y ( y e. A /\ ph ) )
11	3 10	nfxfrd	|- ( T. -> F/ x E! y e. A ph )
12	11	mptru	|- F/ x E! y e. A ph