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Theorem bdsepnft 12887
Description: Closed form of bdsepnf 12888. Version of ax-bdsep 12884 with one disjoint variable condition removed, the other disjoint variable condition replaced by a non-freeness antecedent, and without initial universal quantifier. Use bdsep1 12885 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 12886 . 2 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑎𝜑))
3 nfnf1 1506 . . . 4 𝑏𝑏𝜑
43nfal 1538 . . 3 𝑏𝑥𝑏𝜑
5 nfa1 1504 . . . 4 𝑥𝑥𝑏𝜑
6 nfvd 1492 . . . . 5 (∀𝑥𝑏𝜑 → Ⅎ𝑏 𝑥𝑦)
7 nfv 1491 . . . . . . 7 𝑏 𝑥𝑎
87a1i 9 . . . . . 6 (∀𝑥𝑏𝜑 → Ⅎ𝑏 𝑥𝑎)
9 sp 1471 . . . . . 6 (∀𝑥𝑏𝜑 → Ⅎ𝑏𝜑)
108, 9nfand 1530 . . . . 5 (∀𝑥𝑏𝜑 → Ⅎ𝑏(𝑥𝑎𝜑))
116, 10nfbid 1550 . . . 4 (∀𝑥𝑏𝜑 → Ⅎ𝑏(𝑥𝑦 ↔ (𝑥𝑎𝜑)))
125, 11nfald 1716 . . 3 (∀𝑥𝑏𝜑 → Ⅎ𝑏𝑥(𝑥𝑦 ↔ (𝑥𝑎𝜑)))
13 nfv 1491 . . . . . 6 𝑥 𝑦 = 𝑏
145, 13nfan 1527 . . . . 5 𝑥(∀𝑥𝑏𝜑𝑦 = 𝑏)
15 elequ2 1674 . . . . . . 7 (𝑦 = 𝑏 → (𝑥𝑦𝑥𝑏))
1615adantl 273 . . . . . 6 ((∀𝑥𝑏𝜑𝑦 = 𝑏) → (𝑥𝑦𝑥𝑏))
1716bibi1d 232 . . . . 5 ((∀𝑥𝑏𝜑𝑦 = 𝑏) → ((𝑥𝑦 ↔ (𝑥𝑎𝜑)) ↔ (𝑥𝑏 ↔ (𝑥𝑎𝜑))))
1814, 17albid 1577 . . . 4 ((∀𝑥𝑏𝜑𝑦 = 𝑏) → (∀𝑥(𝑥𝑦 ↔ (𝑥𝑎𝜑)) ↔ ∀𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))))
1918ex 114 . . 3 (∀𝑥𝑏𝜑 → (𝑦 = 𝑏 → (∀𝑥(𝑥𝑦 ↔ (𝑥𝑎𝜑)) ↔ ∀𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑)))))
204, 12, 19cbvexd 1877 . 2 (∀𝑥𝑏𝜑 → (∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑎𝜑)) ↔ ∃𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))))
212, 20mpbii 147 1 (∀𝑥𝑏𝜑 → ∃𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1312  wnf 1419  wex 1451  BOUNDED wbd 12812
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 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-bdsep 12884
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-cleq 2108  df-clel 2111
This theorem is referenced by:  bdsepnf  12888
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