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Theorem nfdisjv 3747
 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 x A 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 3737 . 2 (Disj x A Bz∃*x(x A z B))
2 nfcv 2175 . . . . . 6 yx
3 nfdisjv.1 . . . . . 6 yA
42, 3nfel 2183 . . . . 5 y x A
5 nfdisjv.2 . . . . . 6 yB
65nfcri 2169 . . . . 5 y z B
74, 6nfan 1454 . . . 4 y(x A z B)
87nfmo 1917 . . 3 y∃*x(x A z B)
98nfal 1465 . 2 yz∃*x(x A z B)
101, 9nfxfr 1360 1 yDisj x A B
 Colors of variables: wff set class Syntax hints:   ∧ wa 97  ∀wal 1240  Ⅎwnf 1346   ∈ wcel 1390  ∃*wmo 1898  Ⅎwnfc 2162  Disj wdisj 3735 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rmo 2308  df-disj 3736 This theorem is referenced by: (None)
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