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Theorem bdsepnft 10366
Description: Closed form of bdsepnf 10367. Version of ax-bdsep 10363 with one DV condition removed, the other DV condition replaced by a non-freeness antecedent, and without initial universal quantifier. Use bdsep1 10364 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 10365 . 2 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑎𝜑))
3 nfnf1 1452 . . . 4 𝑏𝑏𝜑
43nfal 1484 . . 3 𝑏𝑥𝑏𝜑
5 nfa1 1450 . . . 4 𝑥𝑥𝑏𝜑
6 nfvd 1438 . . . . 5 (∀𝑥𝑏𝜑 → Ⅎ𝑏 𝑥𝑦)
7 nfv 1437 . . . . . . 7 𝑏 𝑥𝑎
87a1i 9 . . . . . 6 (∀𝑥𝑏𝜑 → Ⅎ𝑏 𝑥𝑎)
9 sp 1417 . . . . . 6 (∀𝑥𝑏𝜑 → Ⅎ𝑏𝜑)
108, 9nfand 1476 . . . . 5 (∀𝑥𝑏𝜑 → Ⅎ𝑏(𝑥𝑎𝜑))
116, 10nfbid 1496 . . . 4 (∀𝑥𝑏𝜑 → Ⅎ𝑏(𝑥𝑦 ↔ (𝑥𝑎𝜑)))
125, 11nfald 1659 . . 3 (∀𝑥𝑏𝜑 → Ⅎ𝑏𝑥(𝑥𝑦 ↔ (𝑥𝑎𝜑)))
13 nfv 1437 . . . . . 6 𝑥 𝑦 = 𝑏
145, 13nfan 1473 . . . . 5 𝑥(∀𝑥𝑏𝜑𝑦 = 𝑏)
15 elequ2 1617 . . . . . . 7 (𝑦 = 𝑏 → (𝑥𝑦𝑥𝑏))
1615adantl 266 . . . . . 6 ((∀𝑥𝑏𝜑𝑦 = 𝑏) → (𝑥𝑦𝑥𝑏))
1716bibi1d 226 . . . . 5 ((∀𝑥𝑏𝜑𝑦 = 𝑏) → ((𝑥𝑦 ↔ (𝑥𝑎𝜑)) ↔ (𝑥𝑏 ↔ (𝑥𝑎𝜑))))
1814, 17albid 1522 . . . 4 ((∀𝑥𝑏𝜑𝑦 = 𝑏) → (∀𝑥(𝑥𝑦 ↔ (𝑥𝑎𝜑)) ↔ ∀𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))))
1918ex 112 . . 3 (∀𝑥𝑏𝜑 → (𝑦 = 𝑏 → (∀𝑥(𝑥𝑦 ↔ (𝑥𝑎𝜑)) ↔ ∀𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑)))))
204, 12, 19cbvexd 1818 . 2 (∀𝑥𝑏𝜑 → (∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑎𝜑)) ↔ ∃𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))))
212, 20mpbii 140 1 (∀𝑥𝑏𝜑 → ∃𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wal 1257  wnf 1365  wex 1397  BOUNDED wbd 10291
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-bdsep 10363
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-cleq 2049  df-clel 2052
This theorem is referenced by:  bdsepnf  10367
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