Theorem nfdisjw 5092

Description: Bound-variable hypothesis builder for disjoint collection. Version of nfdisj 5093 with a disjoint variable condition, which does not require ax-13 2410. (Contributed by Mario Carneiro, 14-Nov-2016.) Avoid ax-13 2410. (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 5082	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑧∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))
2		nftru 1831	. . . . 5 ⊢ Ⅎ𝑥⊤
3		nfdisjw.1	. . . . . . . 8 ⊢ Ⅎ𝑦𝐴
4	3	a1i 11	. . . . . . 7 ⊢ (⊤ → Ⅎ𝑦𝐴)
5	4	nfcrd 2925	. . . . . 6 ⊢ (⊤ → Ⅎ𝑦 𝑥 ∈ 𝐴)
6		nfdisjw.2	. . . . . . . 8 ⊢ Ⅎ𝑦𝐵
7	6	nfcri 2923	. . . . . . 7 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵
8	7	a1i 11	. . . . . 6 ⊢ (⊤ → Ⅎ𝑦 𝑧 ∈ 𝐵)
9	5, 8	nfand 1924	. . . . 5 ⊢ (⊤ → Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))
10	2, 9	nfmodv 2593	. . . 4 ⊢ (⊤ → Ⅎ𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))
11	10	mptru 1574	. . 3 ⊢ Ⅎ𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)
12	11	nfal 2362	. 2 ⊢ Ⅎ𝑦∀𝑧∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)
13	1, 12	nfxfr 1880	1 ⊢ Ⅎ𝑦Disj 𝑥 ∈ 𝐴 𝐵

Syntax hints:   ∧ wa 400  ∀wal 1565  ⊤wtru 1568  Ⅎwnf 1810   ∈ wcel 2149  ∃*wmo 2571  Ⅎwnfc 2916  Disj wdisj 5080

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-10 2182  ax-11 2198  ax-12 2219

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-nf 1811  df-mo 2573  df-clel 2844  df-nfc 2918  df-rmo 3376  df-disj 5081

This theorem is referenced by:  disjxiun  5110