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Theorem bdsepnfALT 13014
Description: Alternate proof of bdsepnf 13013, not using bdsepnft 13012. (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 13011 . 2 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑎𝜑))
3 nfv 1493 . . . . 5 𝑏 𝑥𝑦
4 nfv 1493 . . . . . 6 𝑏 𝑥𝑎
5 bdsepnf.nf . . . . . 6 𝑏𝜑
64, 5nfan 1529 . . . . 5 𝑏(𝑥𝑎𝜑)
73, 6nfbi 1553 . . . 4 𝑏(𝑥𝑦 ↔ (𝑥𝑎𝜑))
87nfal 1540 . . 3 𝑏𝑥(𝑥𝑦 ↔ (𝑥𝑎𝜑))
9 nfv 1493 . . 3 𝑦𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))
10 elequ2 1676 . . . . 5 (𝑦 = 𝑏 → (𝑥𝑦𝑥𝑏))
1110bibi1d 232 . . . 4 (𝑦 = 𝑏 → ((𝑥𝑦 ↔ (𝑥𝑎𝜑)) ↔ (𝑥𝑏 ↔ (𝑥𝑎𝜑))))
1211albidv 1780 . . 3 (𝑦 = 𝑏 → (∀𝑥(𝑥𝑦 ↔ (𝑥𝑎𝜑)) ↔ ∀𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))))
138, 9, 12cbvex 1714 . 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 1314  wnf 1421  wex 1453  BOUNDED wbd 12937
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 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-bdsep 13009
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-cleq 2110  df-clel 2113
This theorem is referenced by: (None)
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