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		nfcvd	|- ( T. -> F/_ y x )
6	1	a1i	|- ( T. -> F/_ y A )
7	5 6	nfeld	|- ( T. -> F/ y x e. A )
8	2	nfcri	|- F/ y z e. B
9	8	a1i	|- ( T. -> F/ y z e. B )
10	7 9	nfand	|- ( T. -> F/ y ( x e. A /\ z e. B ) )
11	4 10	nfmodv	|- ( T. -> F/ y E* x ( x e. A /\ z e. B ) )
12	11	mptru	|- F/ y E* x ( x e. A /\ z e. B )
13	12	nfal	|- F/ y A. z E* x ( x e. A /\ z e. B )
14	3 13	nfxfr	|- F/ y Disj_ x e. A B