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Theorem bdsep2 9340
Description: Version of ax-bdsep 9339 with one DV condition removed and without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.)
Hypothesis
Ref Expression
bdsep2.1 BOUNDED φ
Assertion
Ref Expression
bdsep2 𝑏x(x 𝑏 ↔ (x 𝑎 φ))
Distinct variable groups:   𝑎,𝑏,x   φ,𝑏
Allowed substitution hints:   φ(x,𝑎)

Proof of Theorem bdsep2
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 eleq2 2098 . . . . . 6 (y = 𝑎 → (x yx 𝑎))
21anbi1d 438 . . . . 5 (y = 𝑎 → ((x y φ) ↔ (x 𝑎 φ)))
32bibi2d 221 . . . 4 (y = 𝑎 → ((x 𝑏 ↔ (x y φ)) ↔ (x 𝑏 ↔ (x 𝑎 φ))))
43albidv 1702 . . 3 (y = 𝑎 → (x(x 𝑏 ↔ (x y φ)) ↔ x(x 𝑏 ↔ (x 𝑎 φ))))
54exbidv 1703 . 2 (y = 𝑎 → (𝑏x(x 𝑏 ↔ (x y φ)) ↔ 𝑏x(x 𝑏 ↔ (x 𝑎 φ))))
6 bdsep2.1 . . . 4 BOUNDED φ
76ax-bdsep 9339 . . 3 y𝑏x(x 𝑏 ↔ (x y φ))
87spi 1426 . 2 𝑏x(x 𝑏 ↔ (x y φ))
95, 8chvarv 1809 1 𝑏x(x 𝑏 ↔ (x 𝑎 φ))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98  wal 1240  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-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  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:  bdsepnft  9341  bdsepnfALT  9343  bdzfauscl  9344  bdbm1.3ii  9345  bj-nalset  9346
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