Theorem nfrexdw

Description: Deduction version of nfrexw . (Contributed by Mario Carneiro, 14-Oct-2016) Add disjoint variable condition to avoid ax-13 . See nfrexd for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024)

	Ref	Expression
Hypotheses	nfraldw.1	|- F/ y ph
	nfraldw.2	|- ( ph -> F/_ x A )
	nfraldw.3	|- ( ph -> F/ x ps )
Assertion	nfrexdw	|- ( ph -> F/ x E. y e. A ps )

Proof

Step	Hyp	Ref	Expression
1		nfraldw.1	|- F/ y ph
2		nfraldw.2	|- ( ph -> F/_ x A )
3		nfraldw.3	|- ( ph -> F/ x ps )
4		dfrex2	|- ( E. y e. A ps <-> -. A. y e. A -. ps )
5	3	nfnd	|- ( ph -> F/ x -. ps )
6	1 2 5	nfraldw	|- ( ph -> F/ x A. y e. A -. ps )
7	6	nfnd	|- ( ph -> F/ x -. A. y e. A -. ps )
8	4 7	nfxfrd	|- ( ph -> F/ x E. y e. A ps )