Theorem nfrex

Description: Bound-variable hypothesis builder for restricted quantification. (Contributed by NM, 1-Sep-1999) (Revised by Mario Carneiro, 7-Oct-2016) (Proof shortened by Wolf Lammen, 30-Dec-2019) Add disjoint variable condition to avoid ax-13 . See nfrexg for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024)

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

Proof

Step	Hyp	Ref	Expression
1		nfrex.1	|- F/_ x A
2		nfrex.2	|- F/ x ph
3		nftru	|- F/ y T.
4	1	a1i	|- ( T. -> F/_ x A )
5	2	a1i	|- ( T. -> F/ x ph )
6	3 4 5	nfrexd	|- ( T. -> F/ x E. y e. A ph )
7	6	mptru	|- F/ x E. y e. A ph