Theorem bdsepnft 13922

Description: Closed form of bdsepnf 13923. Version of ax-bdsep 13919 with one disjoint variable condition removed, the other disjoint variable condition replaced by a nonfreeness antecedent, and without initial universal quantifier. Use bdsep1 13920 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 13921	. 2 ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))
3		nfnf1 1537	. . . 4 ⊢ Ⅎ𝑏Ⅎ𝑏𝜑
4	3	nfal 1569	. . 3 ⊢ Ⅎ𝑏∀𝑥Ⅎ𝑏𝜑
5		nfa1 1534	. . . 4 ⊢ Ⅎ𝑥∀𝑥Ⅎ𝑏𝜑
6		nfvd 1522	. . . . 5 ⊢ (∀𝑥Ⅎ𝑏𝜑 → Ⅎ𝑏 𝑥 ∈ 𝑦)
7		nfv 1521	. . . . . . 7 ⊢ Ⅎ𝑏 𝑥 ∈ 𝑎
8	7	a1i 9	. . . . . 6 ⊢ (∀𝑥Ⅎ𝑏𝜑 → Ⅎ𝑏 𝑥 ∈ 𝑎)
9		sp 1504	. . . . . 6 ⊢ (∀𝑥Ⅎ𝑏𝜑 → Ⅎ𝑏𝜑)
10	8, 9	nfand 1561	. . . . 5 ⊢ (∀𝑥Ⅎ𝑏𝜑 → Ⅎ𝑏(𝑥 ∈ 𝑎 ∧ 𝜑))
11	6, 10	nfbid 1581	. . . 4 ⊢ (∀𝑥Ⅎ𝑏𝜑 → Ⅎ𝑏(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)))
12	5, 11	nfald 1753	. . 3 ⊢ (∀𝑥Ⅎ𝑏𝜑 → Ⅎ𝑏∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)))
13		nfv 1521	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 = 𝑏
14	5, 13	nfan 1558	. . . . 5 ⊢ Ⅎ𝑥(∀𝑥Ⅎ𝑏𝜑 ∧ 𝑦 = 𝑏)
15		elequ2 2146	. . . . . . 7 ⊢ (𝑦 = 𝑏 → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑏))
16	15	adantl 275	. . . . . 6 ⊢ ((∀𝑥Ⅎ𝑏𝜑 ∧ 𝑦 = 𝑏) → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑏))
17	16	bibi1d 232	. . . . 5 ⊢ ((∀𝑥Ⅎ𝑏𝜑 ∧ 𝑦 = 𝑏) → ((𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) ↔ (𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))))
18	14, 17	albid 1608	. . . 4 ⊢ ((∀𝑥Ⅎ𝑏𝜑 ∧ 𝑦 = 𝑏) → (∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) ↔ ∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))))
19	18	ex 114	. . 3 ⊢ (∀𝑥Ⅎ𝑏𝜑 → (𝑦 = 𝑏 → (∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) ↔ ∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)))))
20	4, 12, 19	cbvexd 1920	. 2 ⊢ (∀𝑥Ⅎ𝑏𝜑 → (∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) ↔ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))))
21	2, 20	mpbii 147	1 ⊢ (∀𝑥Ⅎ𝑏𝜑 → ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1346  Ⅎwnf 1453  ∃wex 1485  BOUNDED wbd 13847

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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-bdsep 13919

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-cleq 2163  df-clel 2166

This theorem is referenced by:  bdsepnf  13923