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Theorem bdsepnf 15077
Description: Version of ax-bdsep 15073 with one disjoint variable condition removed, the other disjoint variable condition replaced by a nonfreeness hypothesis, and without initial universal quantifier. See also bdsepnfALT 15078. Use bdsep1 15074 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 15076 . 2 (∀𝑥𝑏𝜑 → ∃𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑)))
3 bdsepnf.nf . 2 𝑏𝜑
42, 3mpg 1462 1 𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105  wal 1362  wnf 1471  wex 1503  BOUNDED wbd 15001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-bdsep 15073
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-cleq 2182  df-clel 2185
This theorem is referenced by: (None)
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