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Theorem bdsepnfALT 9873
Description: Alternate proof of bdsepnf 9872, not using bdsepnft 9871. (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 9870 . 2 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑎𝜑))
3 nfv 1421 . . . . 5 𝑏 𝑥𝑦
4 nfv 1421 . . . . . 6 𝑏 𝑥𝑎
5 bdsepnf.nf . . . . . 6 𝑏𝜑
64, 5nfan 1457 . . . . 5 𝑏(𝑥𝑎𝜑)
73, 6nfbi 1481 . . . 4 𝑏(𝑥𝑦 ↔ (𝑥𝑎𝜑))
87nfal 1468 . . 3 𝑏𝑥(𝑥𝑦 ↔ (𝑥𝑎𝜑))
9 nfv 1421 . . 3 𝑦𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))
10 elequ2 1601 . . . . 5 (𝑦 = 𝑏 → (𝑥𝑦𝑥𝑏))
1110bibi1d 222 . . . 4 (𝑦 = 𝑏 → ((𝑥𝑦 ↔ (𝑥𝑎𝜑)) ↔ (𝑥𝑏 ↔ (𝑥𝑎𝜑))))
1211albidv 1705 . . 3 (𝑦 = 𝑏 → (∀𝑥(𝑥𝑦 ↔ (𝑥𝑎𝜑)) ↔ ∀𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))))
138, 9, 12cbvex 1639 . 2 (∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑎𝜑)) ↔ ∃𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑)))
142, 13mpbi 133 1 𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))
Colors of variables: wff set class
Syntax hints:  wa 97  wb 98  wal 1241  wnf 1349  wex 1381  BOUNDED wbd 9796
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 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-bdsep 9868
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-cleq 2033  df-clel 2036
This theorem is referenced by: (None)
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