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Theorem bdsepnfALT 15094
Description: Alternate proof of bdsepnf 15093, not using bdsepnft 15092. (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 15091 . 2 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑎𝜑))
3 nfv 1539 . . . . 5 𝑏 𝑥𝑦
4 nfv 1539 . . . . . 6 𝑏 𝑥𝑎
5 bdsepnf.nf . . . . . 6 𝑏𝜑
64, 5nfan 1576 . . . . 5 𝑏(𝑥𝑎𝜑)
73, 6nfbi 1600 . . . 4 𝑏(𝑥𝑦 ↔ (𝑥𝑎𝜑))
87nfal 1587 . . 3 𝑏𝑥(𝑥𝑦 ↔ (𝑥𝑎𝜑))
9 nfv 1539 . . 3 𝑦𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))
10 elequ2 2165 . . . . 5 (𝑦 = 𝑏 → (𝑥𝑦𝑥𝑏))
1110bibi1d 233 . . . 4 (𝑦 = 𝑏 → ((𝑥𝑦 ↔ (𝑥𝑎𝜑)) ↔ (𝑥𝑏 ↔ (𝑥𝑎𝜑))))
1211albidv 1835 . . 3 (𝑦 = 𝑏 → (∀𝑥(𝑥𝑦 ↔ (𝑥𝑎𝜑)) ↔ ∀𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))))
138, 9, 12cbvex 1767 . 2 (∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑎𝜑)) ↔ ∃𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑)))
142, 13mpbi 145 1 𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105  wal 1362  wnf 1471  wex 1503  BOUNDED wbd 15017
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-bdsep 15089
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-cleq 2182  df-clel 2185
This theorem is referenced by: (None)
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