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Theorem bdsep1 9878
Description: Version of ax-bdsep 9877 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 9877 . 2 𝑎𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))
32spi 1429 1 𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))
Colors of variables: wff set class
Syntax hints:  wa 97  wb 98  wal 1241  wex 1381  BOUNDED wbd 9805
This theorem was proved from axioms:  ax-mp 7  ax-4 1400  ax-bdsep 9877
This theorem is referenced by:  bdsep2  9879  bdzfauscl  9883  bdbm1.3ii  9884  bj-axemptylem  9885  bj-nalset  9888
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