Theorem nfdisjw 5122

Description: Bound-variable hypothesis builder for disjoint collection. Version of nfdisj 5123 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by Mario Carneiro, 14-Nov-2016.) Avoid ax-13 2377. (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 5112	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑧∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))
2		nftru 1804	. . . . 5 ⊢ Ⅎ𝑥⊤
3		nfcvd 2906	. . . . . . 7 ⊢ (⊤ → Ⅎ𝑦𝑥)
4		nfdisjw.1	. . . . . . . 8 ⊢ Ⅎ𝑦𝐴
5	4	a1i 11	. . . . . . 7 ⊢ (⊤ → Ⅎ𝑦𝐴)
6	3, 5	nfeld 2917	. . . . . 6 ⊢ (⊤ → Ⅎ𝑦 𝑥 ∈ 𝐴)
7		nfdisjw.2	. . . . . . . 8 ⊢ Ⅎ𝑦𝐵
8	7	nfcri 2897	. . . . . . 7 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵
9	8	a1i 11	. . . . . 6 ⊢ (⊤ → Ⅎ𝑦 𝑧 ∈ 𝐵)
10	6, 9	nfand 1897	. . . . 5 ⊢ (⊤ → Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))
11	2, 10	nfmodv 2559	. . . 4 ⊢ (⊤ → Ⅎ𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))
12	11	mptru 1547	. . 3 ⊢ Ⅎ𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)
13	12	nfal 2323	. 2 ⊢ Ⅎ𝑦∀𝑧∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)
14	1, 13	nfxfr 1853	1 ⊢ Ⅎ𝑦Disj 𝑥 ∈ 𝐴 𝐵

Syntax hints:   ∧ wa 395  ∀wal 1538  ⊤wtru 1541  Ⅎwnf 1783   ∈ wcel 2108  ∃*wmo 2538  Ⅎwnfc 2890  Disj wdisj 5110

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-mo 2540  df-cleq 2729  df-clel 2816  df-nfc 2892  df-rmo 3380  df-disj 5111

This theorem is referenced by:  disjxiun  5140