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Theorem bdsepnf 12897
Description: Version of ax-bdsep 12893 with one disjoint variable condition removed, the other disjoint variable condition replaced by a non-freeness hypothesis, and without initial universal quantifier. See also bdsepnfALT 12898. Use bdsep1 12894 when sufficient. (Contributed by BJ, 5-Oct-2019.)
Hypotheses
Ref Expression
bdsepnf.nf 𝑏𝜑
bdsepnf.1 BOUNDED 𝜑
Assertion
Ref Expression
bdsepnf 𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))
Distinct variable group:   𝑎,𝑏,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑎,𝑏)

Proof of Theorem bdsepnf
StepHypRef Expression
1 bdsepnf.1 . . 3 BOUNDED 𝜑
21bdsepnft 12896 . 2 (∀𝑥𝑏𝜑 → ∃𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑)))
3 bdsepnf.nf . 2 𝑏𝜑
42, 3mpg 1410 1 𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wal 1312  wnf 1419  wex 1451  BOUNDED wbd 12821
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 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-bdsep 12893
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-cleq 2108  df-clel 2111
This theorem is referenced by: (None)
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