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

Theorem bdsepnfALT 12921
Description: Alternate proof of bdsepnf 12920, not using bdsepnft 12919. (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 12918 . 2 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑎𝜑))
3 nfv 1491 . . . . 5 𝑏 𝑥𝑦
4 nfv 1491 . . . . . 6 𝑏 𝑥𝑎
5 bdsepnf.nf . . . . . 6 𝑏𝜑
64, 5nfan 1527 . . . . 5 𝑏(𝑥𝑎𝜑)
73, 6nfbi 1551 . . . 4 𝑏(𝑥𝑦 ↔ (𝑥𝑎𝜑))
87nfal 1538 . . 3 𝑏𝑥(𝑥𝑦 ↔ (𝑥𝑎𝜑))
9 nfv 1491 . . 3 𝑦𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))
10 elequ2 1674 . . . . 5 (𝑦 = 𝑏 → (𝑥𝑦𝑥𝑏))
1110bibi1d 232 . . . 4 (𝑦 = 𝑏 → ((𝑥𝑦 ↔ (𝑥𝑎𝜑)) ↔ (𝑥𝑏 ↔ (𝑥𝑎𝜑))))
1211albidv 1778 . . 3 (𝑦 = 𝑏 → (∀𝑥(𝑥𝑦 ↔ (𝑥𝑎𝜑)) ↔ ∀𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))))
138, 9, 12cbvex 1712 . 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 1312  wnf 1419  wex 1451  BOUNDED wbd 12844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-bdsep 12916
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-cleq 2108  df-clel 2111
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator