Theorem nfdisjw 5090

Description: Bound-variable hypothesis builder for disjoint collection. Version of nfdisj 5091 with a disjoint variable condition, which does not require ax-13 2406. (Contributed by Mario Carneiro, 14-Nov-2016.) Avoid ax-13 2406. (Revised by GG, 26-Jan-2024.)

Hypotheses

Ref	Expression
nfdisjw.1	⊢ Ⅎ𝑦𝐴
nfdisjw.2	⊢ Ⅎ𝑦𝐵

Assertion

Ref	Expression
nfdisjw	⊢ Ⅎ𝑦Disj 𝑥 ∈ 𝐴 𝐵

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥, 𝑦)   𝐵(𝑥, 𝑦)

Proof of Theorem nfdisjw

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfdisj2 5080	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑧∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))
2		nftru 1837	. . . . 5 ⊢ Ⅎ𝑥⊤
3		nfdisjw.1	. . . . . . . 8 ⊢ Ⅎ𝑦𝐴
4	3	a1i 11	. . . . . . 7 ⊢ (⊤ → Ⅎ𝑦𝐴)
5	4	nfcrd 2921	. . . . . 6 ⊢ (⊤ → Ⅎ𝑦 𝑥 ∈ 𝐴)
6		nfdisjw.2	. . . . . . . 8 ⊢ Ⅎ𝑦𝐵
7	6	nfcri 2919	. . . . . . 7 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵
8	7	a1i 11	. . . . . 6 ⊢ (⊤ → Ⅎ𝑦 𝑧 ∈ 𝐵)
9	5, 8	nfand 1930	. . . . 5 ⊢ (⊤ → Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))
10	2, 9	nfmodv 2589	. . . 4 ⊢ (⊤ → Ⅎ𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))
11	10	mptru 1577	. . 3 ⊢ Ⅎ𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)
12	11	nfal 2358	. 2 ⊢ Ⅎ𝑦∀𝑧∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)
13	1, 12	nfxfr 1886	1 ⊢ Ⅎ𝑦Disj 𝑥 ∈ 𝐴 𝐵

This proof depends on syntax axioms:   ∧ wa 401  ∀wal 1568  ⊤wtru 1571  Ⅎwnf 1816   ∈ wcel 2146  ∃*wmo 2567  Ⅎwnfc 2912  Disj wdisj 5078

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-10 2179  ax-11 2195  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-mo 2569  df-clel 2840  df-nfc 2914  df-rmo 3371  df-disj 5079

This theorem is used by:  disjxiun  5108