Theorem nfdisjw

Description: Bound-variable hypothesis builder for disjoint collection. Version of nfdisj with a disjoint variable condition, which does not require ax-13 . (Contributed by Mario Carneiro, 14-Nov-2016) Avoid ax-13 . (Revised by GG, 26-Jan-2024)

	Ref	Expression
Hypotheses	nfdisjw.1	|- F/_ y A
	nfdisjw.2	|- F/_ y B
Assertion	nfdisjw	|- F/ y Disj_ x e. A B

Proof

Step	Hyp	Ref	Expression
1		nfdisjw.1	|- F/_ y A
2		nfdisjw.2	|- F/_ y B
3		dfdisj2	|- ( Disj_ x e. A B <-> A. z E* x ( x e. A /\ z e. B ) )
4		nftru	|- F/ x T.
5	1	a1i	|- ( T. -> F/_ y A )
6	5	nfcrd	|- ( T. -> F/ y x e. A )
7	2	nfcri	|- F/ y z e. B
8	7	a1i	|- ( T. -> F/ y z e. B )
9	6 8	nfand	|- ( T. -> F/ y ( x e. A /\ z e. B ) )
10	4 9	nfmodv	|- ( T. -> F/ y E* x ( x e. A /\ z e. B ) )
11	10	mptru	|- F/ y E* x ( x e. A /\ z e. B )
12	11	nfal	|- F/ y A. z E* x ( x e. A /\ z e. B )
13	3 12	nfxfr	|- F/ y Disj_ x e. A B