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Theorem bdsepnft 13115
 Description: Closed form of bdsepnf 13116. Version of ax-bdsep 13112 with one disjoint variable condition removed, the other disjoint variable condition replaced by a non-freeness antecedent, and without initial universal quantifier. Use bdsep1 13113 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 13114 . 2
3 nfnf1 1523 . . . 4
43nfal 1555 . . 3
5 nfa1 1521 . . . 4
6 nfvd 1509 . . . . 5
7 nfv 1508 . . . . . . 7
87a1i 9 . . . . . 6
9 sp 1488 . . . . . 6
108, 9nfand 1547 . . . . 5
116, 10nfbid 1567 . . . 4
125, 11nfald 1733 . . 3
13 nfv 1508 . . . . . 6
145, 13nfan 1544 . . . . 5
15 elequ2 1691 . . . . . . 7
1615adantl 275 . . . . . 6
1716bibi1d 232 . . . . 5
1814, 17albid 1594 . . . 4
1918ex 114 . . 3
204, 12, 19cbvexd 1899 . 2
212, 20mpbii 147 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wb 104  wal 1329  wnf 1436  wex 1468  BOUNDED wbd 13040 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-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-bdsep 13112 This theorem depends on definitions:  df-bi 116  df-nf 1437  df-cleq 2132  df-clel 2135 This theorem is referenced by:  bdsepnf  13116
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