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Theorem bdsepnft 10007
 Description: Closed form of bdsepnf 10008. Version of ax-bdsep 10004 with one DV condition removed, the other DV condition replaced by a non-freeness antecedent, and without initial universal quantifier. Use bdsep1 10005 when sufficient. (Contributed by BJ, 19-Oct-2019.)
Hypothesis
Ref Expression
bdsepnft.1 BOUNDED
Assertion
Ref Expression
bdsepnft
Distinct variable group:   ,,
Allowed substitution hints:   (,,)

Proof of Theorem bdsepnft
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 bdsepnft.1 . . 3 BOUNDED
21bdsep2 10006 . 2
3 nfnf1 1436 . . . 4
43nfal 1468 . . 3
5 nfa1 1434 . . . 4
6 nfvd 1422 . . . . 5
7 nfv 1421 . . . . . . 7
87a1i 9 . . . . . 6
9 sp 1401 . . . . . 6
108, 9nfand 1460 . . . . 5
116, 10nfbid 1480 . . . 4
125, 11nfald 1643 . . 3
13 nfv 1421 . . . . . 6
145, 13nfan 1457 . . . . 5
15 elequ2 1601 . . . . . . 7
1615adantl 262 . . . . . 6
1716bibi1d 222 . . . . 5
1814, 17albid 1506 . . . 4
1918ex 108 . . 3
204, 12, 19cbvexd 1802 . 2
212, 20mpbii 136 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97   wb 98  wal 1241  wnf 1349  wex 1381  BOUNDED wbd 9932 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-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-bdsep 10004 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-cleq 2033  df-clel 2036 This theorem is referenced by:  bdsepnf  10008
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