Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bdsep1 GIF version

Theorem bdsep1 11776
Description: Version of ax-bdsep 11775 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 11775 . 2 𝑎𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))
32spi 1474 1 𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103  wal 1287  wex 1426  BOUNDED wbd 11703
This theorem was proved from axioms:  ax-mp 7  ax-4 1445  ax-bdsep 11775
This theorem is referenced by:  bdsep2  11777  bdzfauscl  11781  bdbm1.3ii  11782  bj-axemptylem  11783  bj-nalset  11786
  Copyright terms: Public domain W3C validator