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Theorem bdsepnf 9343
 Description: Version of ax-bdsep 9339 with one DV condition removed, the other DV condition replaced by a non-freeness hypothesis, and without initial universal quantifier. See also bdsepnfALT 9344. Use bdsep1 9340 when sufficient. (Contributed by BJ, 5-Oct-2019.)
Hypotheses
Ref Expression
bdsepnf.nf 𝑏φ
bdsepnf.1 BOUNDED φ
Assertion
Ref Expression
bdsepnf 𝑏x(x 𝑏 ↔ (x 𝑎 φ))
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 9342 . 2 (x𝑏φ𝑏x(x 𝑏 ↔ (x 𝑎 φ)))
3 bdsepnf.nf . 2 𝑏φ
42, 3mpg 1337 1 𝑏x(x 𝑏 ↔ (x 𝑎 φ))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98  ∀wal 1240  Ⅎwnf 1346  ∃wex 1378  BOUNDED wbd 9267 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 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-bdsep 9339 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-cleq 2030  df-clel 2033 This theorem is referenced by: (None)
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