Theorem nfdisjw 5127

Description: Bound-variable hypothesis builder for disjoint collection. Version of nfdisj 5128 with a disjoint variable condition, which does not require ax-13 2375. (Contributed by Mario Carneiro, 14-Nov-2016.) Avoid ax-13 2375. (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 5117	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑧∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))
2		nftru 1801	. . . . 5 ⊢ Ⅎ𝑥⊤
3		nfcvd 2904	. . . . . . 7 ⊢ (⊤ → Ⅎ𝑦𝑥)
4		nfdisjw.1	. . . . . . . 8 ⊢ Ⅎ𝑦𝐴
5	4	a1i 11	. . . . . . 7 ⊢ (⊤ → Ⅎ𝑦𝐴)
6	3, 5	nfeld 2915	. . . . . 6 ⊢ (⊤ → Ⅎ𝑦 𝑥 ∈ 𝐴)
7		nfdisjw.2	. . . . . . . 8 ⊢ Ⅎ𝑦𝐵
8	7	nfcri 2895	. . . . . . 7 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵
9	8	a1i 11	. . . . . 6 ⊢ (⊤ → Ⅎ𝑦 𝑧 ∈ 𝐵)
10	6, 9	nfand 1895	. . . . 5 ⊢ (⊤ → Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))
11	2, 10	nfmodv 2557	. . . 4 ⊢ (⊤ → Ⅎ𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))
12	11	mptru 1544	. . 3 ⊢ Ⅎ𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)
13	12	nfal 2322	. 2 ⊢ Ⅎ𝑦∀𝑧∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)
14	1, 13	nfxfr 1850	1 ⊢ Ⅎ𝑦Disj 𝑥 ∈ 𝐴 𝐵

Syntax hints:   ∧ wa 395  ∀wal 1535  ⊤wtru 1538  Ⅎwnf 1780   ∈ wcel 2106  ∃*wmo 2536  Ⅎwnfc 2888  Disj wdisj 5115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-mo 2538  df-cleq 2727  df-clel 2814  df-nfc 2890  df-rmo 3378  df-disj 5116

This theorem is referenced by:  disjxiun  5145