Theorem nfiotadw

Description: Deduction version of nfiotaw . Version of nfiotad with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 18-Feb-2013) Avoid ax-13 . (Revised by GG, 26-Jan-2024)

	Ref	Expression
Hypotheses	nfiotadw.1	|- F/ y ph
	nfiotadw.2	|- ( ph -> F/ x ps )
Assertion	nfiotadw	|- ( ph -> F/_ x ( iota y ps ) )

Proof

Step	Hyp	Ref	Expression
1		nfiotadw.1	|- F/ y ph
2		nfiotadw.2	|- ( ph -> F/ x ps )
3		dfiota2	|- ( iota y ps ) = U. { z | A. y ( ps <-> y = z ) }
4		nfv	|- F/ z ph
5		nfvd	|- ( ph -> F/ x y = z )
6	2 5	nfbid	|- ( ph -> F/ x ( ps <-> y = z ) )
7	1 6	nfald	|- ( ph -> F/ x A. y ( ps <-> y = z ) )
8	4 7	nfabdw	|- ( ph -> F/_ x { z | A. y ( ps <-> y = z ) } )
9	8	nfunid	|- ( ph -> F/_ x U. { z | A. y ( ps <-> y = z ) } )
10	3 9	nfcxfrd	|- ( ph -> F/_ x ( iota y ps ) )