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Theorem bdsep1 10392
Description: Version of ax-bdsep 10391 without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.)
Hypothesis
Ref Expression
bdsep1.1 BOUNDED 𝜑
Assertion
Ref Expression
bdsep1 𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))
Distinct variable groups:   𝑎,𝑏,𝑥   𝜑,𝑎,𝑏
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem bdsep1
StepHypRef Expression
1 bdsep1.1 . . 3 BOUNDED 𝜑
21ax-bdsep 10391 . 2 𝑎𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))
32spi 1445 1 𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102  wal 1257  wex 1397  BOUNDED wbd 10319
This theorem was proved from axioms:  ax-mp 7  ax-4 1416  ax-bdsep 10391
This theorem is referenced by:  bdsep2  10393  bdzfauscl  10397  bdbm1.3ii  10398  bj-axemptylem  10399  bj-nalset  10402
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