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Theorem nfdisjv 3921
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 3911 . 2 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑧∃*𝑥(𝑥𝐴𝑧𝐵))
2 nfcv 2281 . . . . . 6 𝑦𝑥
3 nfdisjv.1 . . . . . 6 𝑦𝐴
42, 3nfel 2290 . . . . 5 𝑦 𝑥𝐴
5 nfdisjv.2 . . . . . 6 𝑦𝐵
65nfcri 2275 . . . . 5 𝑦 𝑧𝐵
74, 6nfan 1544 . . . 4 𝑦(𝑥𝐴𝑧𝐵)
87nfmo 2019 . . 3 𝑦∃*𝑥(𝑥𝐴𝑧𝐵)
98nfal 1555 . 2 𝑦𝑧∃*𝑥(𝑥𝐴𝑧𝐵)
101, 9nfxfr 1450 1 𝑦Disj 𝑥𝐴 𝐵
Colors of variables: wff set class
Syntax hints:  wa 103  wal 1329  wnf 1436  wcel 1480  ∃*wmo 2000  wnfc 2268  Disj wdisj 3909
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rmo 2424  df-disj 3910
This theorem is referenced by: (None)
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