Theorem bdsep2 16021

Description: Version of ax-bdsep 16019 with one disjoint variable condition removed and without initial universal quantifier. Use bdsep1 16020 when sufficient. (Contributed by BJ, 5-Oct-2019.)

Hypothesis

Ref	Expression
bdsep2.1	⊢ BOUNDED 𝜑

Assertion

Ref	Expression
bdsep2	⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))

Distinct variable groups:   𝑎,𝑏,𝑥   𝜑,𝑏

Allowed substitution hints:   𝜑(𝑥,𝑎)

Proof of Theorem bdsep2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2271	. . . . . 6 ⊢ (𝑦 = 𝑎 → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑎))
2	1	anbi1d 465	. . . . 5 ⊢ (𝑦 = 𝑎 → ((𝑥 ∈ 𝑦 ∧ 𝜑) ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)))
3	2	bibi2d 232	. . . 4 ⊢ (𝑦 = 𝑎 → ((𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑦 ∧ 𝜑)) ↔ (𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))))
4	3	albidv 1848	. . 3 ⊢ (𝑦 = 𝑎 → (∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑦 ∧ 𝜑)) ↔ ∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))))
5	4	exbidv 1849	. 2 ⊢ (𝑦 = 𝑎 → (∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑦 ∧ 𝜑)) ↔ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))))
6		bdsep2.1	. . 3 ⊢ BOUNDED 𝜑
7	6	bdsep1 16020	. 2 ⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑦 ∧ 𝜑))
8	5, 7	chvarv 1966	1 ⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))

Syntax hints:   ∧ wa 104   ↔ wb 105  ∀wal 1371  ∃wex 1516  BOUNDED wbd 15947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-ext 2189  ax-bdsep 16019

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-cleq 2200  df-clel 2203

This theorem is referenced by:  bdsepnft  16022  bdsepnfALT  16024