Theorem nfdisjw 5065

Description: Bound-variable hypothesis builder for disjoint collection. Version of nfdisj 5066 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 5055	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑧∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))
2		nftru 1806	. . . . 5 ⊢ Ⅎ𝑥⊤
3		nfdisjw.1	. . . . . . . 8 ⊢ Ⅎ𝑦𝐴
4	3	a1i 11	. . . . . . 7 ⊢ (⊤ → Ⅎ𝑦𝐴)
5	4	nfcrd 2893	. . . . . 6 ⊢ (⊤ → Ⅎ𝑦 𝑥 ∈ 𝐴)
6		nfdisjw.2	. . . . . . . 8 ⊢ Ⅎ𝑦𝐵
7	6	nfcri 2891	. . . . . . 7 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵
8	7	a1i 11	. . . . . 6 ⊢ (⊤ → Ⅎ𝑦 𝑧 ∈ 𝐵)
9	5, 8	nfand 1899	. . . . 5 ⊢ (⊤ → Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))
10	2, 9	nfmodv 2560	. . . 4 ⊢ (⊤ → Ⅎ𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))
11	10	mptru 1549	. . 3 ⊢ Ⅎ𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)
12	11	nfal 2329	. 2 ⊢ Ⅎ𝑦∀𝑧∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)
13	1, 12	nfxfr 1855	1 ⊢ Ⅎ𝑦Disj 𝑥 ∈ 𝐴 𝐵

Syntax hints:   ∧ wa 395  ∀wal 1540  ⊤wtru 1543  Ⅎwnf 1785   ∈ wcel 2114  ∃*wmo 2538  Ⅎwnfc 2884  Disj wdisj 5053

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-10 2147  ax-11 2163  ax-12 2185

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-mo 2540  df-clel 2812  df-nfc 2886  df-rmo 3343  df-disj 5054

This theorem is referenced by:  disjxiun  5083