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 Gino Giotto, 26-Jan-2024)

	Ref	Expression
Hypotheses	nfdisjw.1	⊢ Ⅎ 𝑦 𝐴
	nfdisjw.2	⊢ Ⅎ 𝑦 𝐵
Assertion	nfdisjw	⊢ Ⅎ 𝑦 Disj 𝑥 ∈ 𝐴 𝐵

Proof

Step	Hyp	Ref	Expression
1		nfdisjw.1	⊢ Ⅎ 𝑦 𝐴
2		nfdisjw.2	⊢ Ⅎ 𝑦 𝐵
3		dfdisj2	⊢ ( Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀ 𝑧 ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) )
4		nftru	⊢ Ⅎ 𝑥 ⊤
5		nfcvd	⊢ ( ⊤ → Ⅎ 𝑦 𝑥 )
6	1	a1i	⊢ ( ⊤ → Ⅎ 𝑦 𝐴 )
7	5 6	nfeld	⊢ ( ⊤ → Ⅎ 𝑦 𝑥 ∈ 𝐴 )
8	2	nfcri	⊢ Ⅎ 𝑦 𝑧 ∈ 𝐵
9	8	a1i	⊢ ( ⊤ → Ⅎ 𝑦 𝑧 ∈ 𝐵 )
10	7 9	nfand	⊢ ( ⊤ → Ⅎ 𝑦 ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) )
11	4 10	nfmodv	⊢ ( ⊤ → Ⅎ 𝑦 ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) )
12	11	mptru	⊢ Ⅎ 𝑦 ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 )
13	12	nfal	⊢ Ⅎ 𝑦 ∀ 𝑧 ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 )
14	3 13	nfxfr	⊢ Ⅎ 𝑦 Disj 𝑥 ∈ 𝐴 𝐵