Theorem nfdisjw 5088

Description: Bound-variable hypothesis builder for disjoint collection. Version of nfdisj 5089 with a disjoint variable condition, which does not require ax-13 2404. (Contributed by Mario Carneiro, 14-Nov-2016.) Avoid ax-13 2404. (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 5078	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑧∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))
2		nftru 1834	. . . . 5 ⊢ Ⅎ𝑥⊤
3		nfdisjw.1	. . . . . . . 8 ⊢ Ⅎ𝑦𝐴
4	3	a1i 11	. . . . . . 7 ⊢ (⊤ → Ⅎ𝑦𝐴)
5	4	nfcrd 2919	. . . . . 6 ⊢ (⊤ → Ⅎ𝑦 𝑥 ∈ 𝐴)
6		nfdisjw.2	. . . . . . . 8 ⊢ Ⅎ𝑦𝐵
7	6	nfcri 2917	. . . . . . 7 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵
8	7	a1i 11	. . . . . 6 ⊢ (⊤ → Ⅎ𝑦 𝑧 ∈ 𝐵)
9	5, 8	nfand 1927	. . . . 5 ⊢ (⊤ → Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))
10	2, 9	nfmodv 2587	. . . 4 ⊢ (⊤ → Ⅎ𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))
11	10	mptru 1577	. . 3 ⊢ Ⅎ𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)
12	11	nfal 2356	. 2 ⊢ Ⅎ𝑦∀𝑧∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)
13	1, 12	nfxfr 1883	1 ⊢ Ⅎ𝑦Disj 𝑥 ∈ 𝐴 𝐵

Syntax hints:   ∧ wa 400  ∀wal 1568  ⊤wtru 1571  Ⅎwnf 1813   ∈ wcel 2143  ∃*wmo 2565  Ⅎwnfc 2910  Disj wdisj 5076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-10 2176  ax-11 2192  ax-12 2213

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-nf 1814  df-mo 2567  df-clel 2838  df-nfc 2912  df-rmo 3369  df-disj 5077

This theorem is referenced by:  disjxiun  5106