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Theorem bdsepnft 9321
Description: Closed form of bdsepnf 9322. Version of ax-bdsep 9319 with one DV condition removed, the other DV condition replaced by a non-freeness antecedent, and without initial universal quantifier. (Contributed by BJ, 19-Oct-2019.)
Hypothesis
Ref Expression
bdsepnft.1 BOUNDED φ
Assertion
Ref Expression
bdsepnft (x𝑏φ𝑏x(x 𝑏 ↔ (x 𝑎 φ)))
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 9320 . 2 yx(x y ↔ (x 𝑎 φ))
3 nfnf1 1433 . . . 4 𝑏𝑏φ
43nfal 1465 . . 3 𝑏x𝑏φ
5 nfa1 1431 . . . 4 xx𝑏φ
6 nfvd 1419 . . . . 5 (x𝑏φ → Ⅎ𝑏 x y)
7 nfv 1418 . . . . . . 7 𝑏 x 𝑎
87a1i 9 . . . . . 6 (x𝑏φ → Ⅎ𝑏 x 𝑎)
9 sp 1398 . . . . . 6 (x𝑏φ → Ⅎ𝑏φ)
108, 9nfand 1457 . . . . 5 (x𝑏φ → Ⅎ𝑏(x 𝑎 φ))
116, 10nfbid 1477 . . . 4 (x𝑏φ → Ⅎ𝑏(x y ↔ (x 𝑎 φ)))
125, 11nfald 1640 . . 3 (x𝑏φ → Ⅎ𝑏x(x y ↔ (x 𝑎 φ)))
13 nfv 1418 . . . . . 6 x y = 𝑏
145, 13nfan 1454 . . . . 5 x(x𝑏φ y = 𝑏)
15 elequ2 1598 . . . . . . 7 (y = 𝑏 → (x yx 𝑏))
1615adantl 262 . . . . . 6 ((x𝑏φ y = 𝑏) → (x yx 𝑏))
1716bibi1d 222 . . . . 5 ((x𝑏φ y = 𝑏) → ((x y ↔ (x 𝑎 φ)) ↔ (x 𝑏 ↔ (x 𝑎 φ))))
1814, 17albid 1503 . . . 4 ((x𝑏φ y = 𝑏) → (x(x y ↔ (x 𝑎 φ)) ↔ x(x 𝑏 ↔ (x 𝑎 φ))))
1918ex 108 . . 3 (x𝑏φ → (y = 𝑏 → (x(x y ↔ (x 𝑎 φ)) ↔ x(x 𝑏 ↔ (x 𝑎 φ)))))
204, 12, 19cbvexd 1799 . 2 (x𝑏φ → (yx(x y ↔ (x 𝑎 φ)) ↔ 𝑏x(x 𝑏 ↔ (x 𝑎 φ))))
212, 20mpbii 136 1 (x𝑏φ𝑏x(x 𝑏 ↔ (x 𝑎 φ)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240  wnf 1346  wex 1378  BOUNDED wbd 9247
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 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-bdsep 9319
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-cleq 2030  df-clel 2033
This theorem is referenced by:  bdsepnf  9322
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