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Theorem nfdisjv 3888
Description: Bound-variable hypothesis builder for disjoint collection. (Contributed by Jim Kingdon, 19-Aug-2018.)
Hypotheses
Ref Expression
nfdisjv.1 𝑦𝐴
nfdisjv.2 𝑦𝐵
Assertion
Ref Expression
nfdisjv 𝑦Disj 𝑥𝐴 𝐵
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem nfdisjv
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dfdisj2 3878 . 2 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑧∃*𝑥(𝑥𝐴𝑧𝐵))
2 nfcv 2258 . . . . . 6 𝑦𝑥
3 nfdisjv.1 . . . . . 6 𝑦𝐴
42, 3nfel 2267 . . . . 5 𝑦 𝑥𝐴
5 nfdisjv.2 . . . . . 6 𝑦𝐵
65nfcri 2252 . . . . 5 𝑦 𝑧𝐵
74, 6nfan 1529 . . . 4 𝑦(𝑥𝐴𝑧𝐵)
87nfmo 1997 . . 3 𝑦∃*𝑥(𝑥𝐴𝑧𝐵)
98nfal 1540 . 2 𝑦𝑧∃*𝑥(𝑥𝐴𝑧𝐵)
101, 9nfxfr 1435 1 𝑦Disj 𝑥𝐴 𝐵
Colors of variables: wff set class
Syntax hints:  wa 103  wal 1314  wnf 1421  wcel 1465  ∃*wmo 1978  wnfc 2245  Disj wdisj 3876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rmo 2401  df-disj 3877
This theorem is referenced by: (None)
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