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Theorem nfdisjv 3784
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 3774 . 2 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑧∃*𝑥(𝑥𝐴𝑧𝐵))
2 nfcv 2194 . . . . . 6 𝑦𝑥
3 nfdisjv.1 . . . . . 6 𝑦𝐴
42, 3nfel 2202 . . . . 5 𝑦 𝑥𝐴
5 nfdisjv.2 . . . . . 6 𝑦𝐵
65nfcri 2188 . . . . 5 𝑦 𝑧𝐵
74, 6nfan 1473 . . . 4 𝑦(𝑥𝐴𝑧𝐵)
87nfmo 1936 . . 3 𝑦∃*𝑥(𝑥𝐴𝑧𝐵)
98nfal 1484 . 2 𝑦𝑧∃*𝑥(𝑥𝐴𝑧𝐵)
101, 9nfxfr 1379 1 𝑦Disj 𝑥𝐴 𝐵
Colors of variables: wff set class
Syntax hints:  wa 101  wal 1257  wnf 1365  wcel 1409  ∃*wmo 1917  wnfc 2181  Disj wdisj 3772
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rmo 2331  df-disj 3773
This theorem is referenced by: (None)
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