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Theorem bdsepnf 11779
Description: Version of ax-bdsep 11775 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 11780. Use bdsep1 11776 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 11778 . 2 (∀𝑥𝑏𝜑 → ∃𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑)))
3 bdsepnf.nf . 2 𝑏𝜑
42, 3mpg 1385 1 𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103  wal 1287  wnf 1394  wex 1426  BOUNDED wbd 11703
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-bdsep 11775
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-cleq 2081  df-clel 2084
This theorem is referenced by: (None)
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