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Theorem nfdisjw 5125
Description: Bound-variable hypothesis builder for disjoint collection. Version of nfdisj 5126 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by Mario Carneiro, 14-Nov-2016.) Avoid ax-13 2370. (Revised by Gino Giotto, 26-Jan-2024.)
Hypotheses
Ref Expression
nfdisjw.1 𝑦𝐴
nfdisjw.2 𝑦𝐵
Assertion
Ref Expression
nfdisjw 𝑦Disj 𝑥𝐴 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.
StepHypRef Expression
1 dfdisj2 5115 . 2 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑧∃*𝑥(𝑥𝐴𝑧𝐵))
2 nftru 1805 . . . . 5 𝑥
3 nfcvd 2903 . . . . . . 7 (⊤ → 𝑦𝑥)
4 nfdisjw.1 . . . . . . . 8 𝑦𝐴
54a1i 11 . . . . . . 7 (⊤ → 𝑦𝐴)
63, 5nfeld 2913 . . . . . 6 (⊤ → Ⅎ𝑦 𝑥𝐴)
7 nfdisjw.2 . . . . . . . 8 𝑦𝐵
87nfcri 2889 . . . . . . 7 𝑦 𝑧𝐵
98a1i 11 . . . . . 6 (⊤ → Ⅎ𝑦 𝑧𝐵)
106, 9nfand 1899 . . . . 5 (⊤ → Ⅎ𝑦(𝑥𝐴𝑧𝐵))
112, 10nfmodv 2552 . . . 4 (⊤ → Ⅎ𝑦∃*𝑥(𝑥𝐴𝑧𝐵))
1211mptru 1547 . . 3 𝑦∃*𝑥(𝑥𝐴𝑧𝐵)
1312nfal 2315 . 2 𝑦𝑧∃*𝑥(𝑥𝐴𝑧𝐵)
141, 13nfxfr 1854 1 𝑦Disj 𝑥𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  wa 395  wal 1538  wtru 1541  wnf 1784  wcel 2105  ∃*wmo 2531  wnfc 2882  Disj wdisj 5113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-mo 2533  df-cleq 2723  df-clel 2809  df-nfc 2884  df-rmo 3375  df-disj 5114
This theorem is referenced by:  disjxiun  5145
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