Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bdsep2 Structured version   GIF version

Theorem bdsep2 7112
Description: Version of ax-bdsep 7111 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 2083 . . . . . 6 (y = 𝑎 → (x yx 𝑎))
21anbi1d 441 . . . . 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 1687 . . 3 (y = 𝑎 → (x(x 𝑏 ↔ (x y φ)) ↔ x(x 𝑏 ↔ (x 𝑎 φ))))
54exbidv 1688 . 2 (y = 𝑎 → (𝑏x(x 𝑏 ↔ (x y φ)) ↔ 𝑏x(x 𝑏 ↔ (x 𝑎 φ))))
6 bdsep2.1 . . . 4 BOUNDED φ
76ax-bdsep 7111 . . 3 y𝑏x(x 𝑏 ↔ (x y φ))
87spi 1411 . 2 𝑏x(x 𝑏 ↔ (x y φ))
95, 8chvarv 1794 1 𝑏x(x 𝑏 ↔ (x 𝑎 φ))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98  wal 1226  wex 1362  BOUNDED wbd 7039
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 1316  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-ext 2004  ax-bdsep 7111
This theorem depends on definitions:  df-bi 110  df-nf 1330  df-cleq 2015  df-clel 2018
This theorem is referenced by:  bdsepnft  7113  bdsepnfALT  7115  bdzfauscl  7116  bdbm1.3ii  7117  bj-nalset  7118
  Copyright terms: Public domain W3C validator