Theorem bdsepnft 10007

Description: Closed form of bdsepnf 10008. Version of ax-bdsep 10004 with one DV condition removed, the other DV condition replaced by a non-freeness antecedent, and without initial universal quantifier. Use bdsep1 10005 when sufficient. (Contributed by BJ, 19-Oct-2019.)

Hypothesis

Ref	Expression
bdsepnft.1	⊢ BOUNDED 𝜑

Assertion

Ref	Expression
bdsepnft	⊢ (∀𝑥Ⅎ𝑏𝜑 → ∃𝑏∀𝑥(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.

Step	Hyp	Ref	Expression
1		bdsepnft.1	. . 3 ⊢ BOUNDED 𝜑
2	1	bdsep2 10006	. 2 ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))
3		nfnf1 1436	. . . 4 ⊢ Ⅎ𝑏Ⅎ𝑏𝜑
4	3	nfal 1468	. . 3 ⊢ Ⅎ𝑏∀𝑥Ⅎ𝑏𝜑
5		nfa1 1434	. . . 4 ⊢ Ⅎ𝑥∀𝑥Ⅎ𝑏𝜑
6		nfvd 1422	. . . . 5 ⊢ (∀𝑥Ⅎ𝑏𝜑 → Ⅎ𝑏 𝑥 ∈ 𝑦)
7		nfv 1421	. . . . . . 7 ⊢ Ⅎ𝑏 𝑥 ∈ 𝑎
8	7	a1i 9	. . . . . 6 ⊢ (∀𝑥Ⅎ𝑏𝜑 → Ⅎ𝑏 𝑥 ∈ 𝑎)
9		sp 1401	. . . . . 6 ⊢ (∀𝑥Ⅎ𝑏𝜑 → Ⅎ𝑏𝜑)
10	8, 9	nfand 1460	. . . . 5 ⊢ (∀𝑥Ⅎ𝑏𝜑 → Ⅎ𝑏(𝑥 ∈ 𝑎 ∧ 𝜑))
11	6, 10	nfbid 1480	. . . 4 ⊢ (∀𝑥Ⅎ𝑏𝜑 → Ⅎ𝑏(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)))
12	5, 11	nfald 1643	. . 3 ⊢ (∀𝑥Ⅎ𝑏𝜑 → Ⅎ𝑏∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)))
13		nfv 1421	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 = 𝑏
14	5, 13	nfan 1457	. . . . 5 ⊢ Ⅎ𝑥(∀𝑥Ⅎ𝑏𝜑 ∧ 𝑦 = 𝑏)
15		elequ2 1601	. . . . . . 7 ⊢ (𝑦 = 𝑏 → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑏))
16	15	adantl 262	. . . . . 6 ⊢ ((∀𝑥Ⅎ𝑏𝜑 ∧ 𝑦 = 𝑏) → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑏))
17	16	bibi1d 222	. . . . 5 ⊢ ((∀𝑥Ⅎ𝑏𝜑 ∧ 𝑦 = 𝑏) → ((𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) ↔ (𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))))
18	14, 17	albid 1506	. . . 4 ⊢ ((∀𝑥Ⅎ𝑏𝜑 ∧ 𝑦 = 𝑏) → (∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) ↔ ∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))))
19	18	ex 108	. . 3 ⊢ (∀𝑥Ⅎ𝑏𝜑 → (𝑦 = 𝑏 → (∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) ↔ ∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)))))
20	4, 12, 19	cbvexd 1802	. 2 ⊢ (∀𝑥Ⅎ𝑏𝜑 → (∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) ↔ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))))
21	2, 20	mpbii 136	1 ⊢ (∀𝑥Ⅎ𝑏𝜑 → ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1241  Ⅎwnf 1349  ∃wex 1381  BOUNDED wbd 9932

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 10004

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-cleq 2033  df-clel 2036

This theorem is referenced by:  bdsepnf  10008