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Theorem nfdisjw 5011
 Description: Bound-variable hypothesis builder for disjoint collection. Version of nfdisj 5012 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by Mario Carneiro, 14-Nov-2016.) (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 5001 . 2 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑧∃*𝑥(𝑥𝐴𝑧𝐵))
2 nftru 1806 . . . . 5 𝑥
3 nfcvd 2956 . . . . . . 7 (⊤ → 𝑦𝑥)
4 nfdisjw.1 . . . . . . . 8 𝑦𝐴
54a1i 11 . . . . . . 7 (⊤ → 𝑦𝐴)
63, 5nfeld 2966 . . . . . 6 (⊤ → Ⅎ𝑦 𝑥𝐴)
7 nfdisjw.2 . . . . . . . 8 𝑦𝐵
87nfcri 2943 . . . . . . 7 𝑦 𝑧𝐵
98a1i 11 . . . . . 6 (⊤ → Ⅎ𝑦 𝑧𝐵)
106, 9nfand 1898 . . . . 5 (⊤ → Ⅎ𝑦(𝑥𝐴𝑧𝐵))
112, 10nfmodv 2618 . . . 4 (⊤ → Ⅎ𝑦∃*𝑥(𝑥𝐴𝑧𝐵))
1211mptru 1545 . . 3 𝑦∃*𝑥(𝑥𝐴𝑧𝐵)
1312nfal 2331 . 2 𝑦𝑧∃*𝑥(𝑥𝐴𝑧𝐵)
141, 13nfxfr 1854 1 𝑦Disj 𝑥𝐴 𝐵
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399  ∀wal 1536  ⊤wtru 1539  Ⅎwnf 1785   ∈ wcel 2111  ∃*wmo 2596  Ⅎwnfc 2936  Disj wdisj 4999 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-mo 2598  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rmo 3114  df-disj 5000 This theorem is referenced by:  disjxiun  5031
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