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

Theorem bdsepnfALT 12492
 Description: Alternate proof of bdsepnf 12491, not using bdsepnft 12490. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
bdsepnf.nf 𝑏𝜑
bdsepnf.1 BOUNDED 𝜑
Assertion
Ref Expression
bdsepnfALT 𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))
Distinct variable group:   𝑎,𝑏,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑎,𝑏)

Proof of Theorem bdsepnfALT
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 bdsepnf.1 . . 3 BOUNDED 𝜑
21bdsep2 12489 . 2 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑎𝜑))
3 nfv 1473 . . . . 5 𝑏 𝑥𝑦
4 nfv 1473 . . . . . 6 𝑏 𝑥𝑎
5 bdsepnf.nf . . . . . 6 𝑏𝜑
64, 5nfan 1509 . . . . 5 𝑏(𝑥𝑎𝜑)
73, 6nfbi 1533 . . . 4 𝑏(𝑥𝑦 ↔ (𝑥𝑎𝜑))
87nfal 1520 . . 3 𝑏𝑥(𝑥𝑦 ↔ (𝑥𝑎𝜑))
9 nfv 1473 . . 3 𝑦𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))
10 elequ2 1655 . . . . 5 (𝑦 = 𝑏 → (𝑥𝑦𝑥𝑏))
1110bibi1d 232 . . . 4 (𝑦 = 𝑏 → ((𝑥𝑦 ↔ (𝑥𝑎𝜑)) ↔ (𝑥𝑏 ↔ (𝑥𝑎𝜑))))
1211albidv 1759 . . 3 (𝑦 = 𝑏 → (∀𝑥(𝑥𝑦 ↔ (𝑥𝑎𝜑)) ↔ ∀𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))))
138, 9, 12cbvex 1693 . 2 (∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑎𝜑)) ↔ ∃𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑)))
142, 13mpbi 144 1 𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   ↔ wb 104  ∀wal 1294  Ⅎwnf 1401  ∃wex 1433  BOUNDED wbd 12415 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-bdsep 12487 This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-cleq 2088  df-clel 2091 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator