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Theorem nfdisjw 5122
Description: Bound-variable hypothesis builder for disjoint collection. Version of nfdisj 5123 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 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 5112 . 2 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑧∃*𝑥(𝑥𝐴𝑧𝐵))
2 nftru 1804 . . . . 5 𝑥
3 nfcvd 2906 . . . . . . 7 (⊤ → 𝑦𝑥)
4 nfdisjw.1 . . . . . . . 8 𝑦𝐴
54a1i 11 . . . . . . 7 (⊤ → 𝑦𝐴)
63, 5nfeld 2917 . . . . . 6 (⊤ → Ⅎ𝑦 𝑥𝐴)
7 nfdisjw.2 . . . . . . . 8 𝑦𝐵
87nfcri 2897 . . . . . . 7 𝑦 𝑧𝐵
98a1i 11 . . . . . 6 (⊤ → Ⅎ𝑦 𝑧𝐵)
106, 9nfand 1897 . . . . 5 (⊤ → Ⅎ𝑦(𝑥𝐴𝑧𝐵))
112, 10nfmodv 2559 . . . 4 (⊤ → Ⅎ𝑦∃*𝑥(𝑥𝐴𝑧𝐵))
1211mptru 1547 . . 3 𝑦∃*𝑥(𝑥𝐴𝑧𝐵)
1312nfal 2323 . 2 𝑦𝑧∃*𝑥(𝑥𝐴𝑧𝐵)
141, 13nfxfr 1853 1 𝑦Disj 𝑥𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  wa 395  wal 1538  wtru 1541  wnf 1783  wcel 2108  ∃*wmo 2538  wnfc 2890  Disj wdisj 5110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-mo 2540  df-cleq 2729  df-clel 2816  df-nfc 2892  df-rmo 3380  df-disj 5111
This theorem is referenced by:  disjxiun  5140
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