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Theorem bdsepnft 9981
Description: Closed form of bdsepnf 9982. Version of ax-bdsep 9978 with one DV condition removed, the other DV condition replaced by a non-freeness antecedent, and without initial universal quantifier. Use bdsep1 9979 when sufficient. (Contributed by BJ, 19-Oct-2019.)
Hypothesis
Ref Expression
bdsepnft.1 BOUNDED 𝜑
Assertion
Ref Expression
bdsepnft (∀𝑥𝑏𝜑 → ∃𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑)))
Distinct variable group:   𝑎,𝑏,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑎,𝑏)

Proof of Theorem bdsepnft
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 bdsepnft.1 . . 3 BOUNDED 𝜑
21bdsep2 9980 . 2 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑎𝜑))
3 nfnf1 1436 . . . 4 𝑏𝑏𝜑
43nfal 1468 . . 3 𝑏𝑥𝑏𝜑
5 nfa1 1434 . . . 4 𝑥𝑥𝑏𝜑
6 nfvd 1422 . . . . 5 (∀𝑥𝑏𝜑 → Ⅎ𝑏 𝑥𝑦)
7 nfv 1421 . . . . . . 7 𝑏 𝑥𝑎
87a1i 9 . . . . . 6 (∀𝑥𝑏𝜑 → Ⅎ𝑏 𝑥𝑎)
9 sp 1401 . . . . . 6 (∀𝑥𝑏𝜑 → Ⅎ𝑏𝜑)
108, 9nfand 1460 . . . . 5 (∀𝑥𝑏𝜑 → Ⅎ𝑏(𝑥𝑎𝜑))
116, 10nfbid 1480 . . . 4 (∀𝑥𝑏𝜑 → Ⅎ𝑏(𝑥𝑦 ↔ (𝑥𝑎𝜑)))
125, 11nfald 1643 . . 3 (∀𝑥𝑏𝜑 → Ⅎ𝑏𝑥(𝑥𝑦 ↔ (𝑥𝑎𝜑)))
13 nfv 1421 . . . . . 6 𝑥 𝑦 = 𝑏
145, 13nfan 1457 . . . . 5 𝑥(∀𝑥𝑏𝜑𝑦 = 𝑏)
15 elequ2 1601 . . . . . . 7 (𝑦 = 𝑏 → (𝑥𝑦𝑥𝑏))
1615adantl 262 . . . . . 6 ((∀𝑥𝑏𝜑𝑦 = 𝑏) → (𝑥𝑦𝑥𝑏))
1716bibi1d 222 . . . . 5 ((∀𝑥𝑏𝜑𝑦 = 𝑏) → ((𝑥𝑦 ↔ (𝑥𝑎𝜑)) ↔ (𝑥𝑏 ↔ (𝑥𝑎𝜑))))
1814, 17albid 1506 . . . 4 ((∀𝑥𝑏𝜑𝑦 = 𝑏) → (∀𝑥(𝑥𝑦 ↔ (𝑥𝑎𝜑)) ↔ ∀𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))))
1918ex 108 . . 3 (∀𝑥𝑏𝜑 → (𝑦 = 𝑏 → (∀𝑥(𝑥𝑦 ↔ (𝑥𝑎𝜑)) ↔ ∀𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑)))))
204, 12, 19cbvexd 1802 . 2 (∀𝑥𝑏𝜑 → (∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑎𝜑)) ↔ ∃𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))))
212, 20mpbii 136 1 (∀𝑥𝑏𝜑 → ∃𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98  wal 1241  wnf 1349  wex 1381  BOUNDED wbd 9906
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 9978
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-cleq 2033  df-clel 2036
This theorem is referenced by:  bdsepnf  9982
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