Theorem bdsep1 13083

Description: Version of ax-bdsep 13082 without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.)

Hypothesis

Ref	Expression
bdsep1.1	⊢ BOUNDED 𝜑

Assertion

Ref	Expression
bdsep1	⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))

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

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem bdsep1

Step	Hyp	Ref	Expression
1		bdsep1.1	. . 3 ⊢ BOUNDED 𝜑
2	1	ax-bdsep 13082	. 2 ⊢ ∀𝑎∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))
3	2	spi 1516	1 ⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))

Syntax hints:   ∧ wa 103   ↔ wb 104  ∀wal 1329  ∃wex 1468  BOUNDED wbd 13010

This theorem was proved from axioms:  ax-mp 5  ax-4 1487  ax-bdsep 13082

This theorem is referenced by:  bdsep2  13084  bdzfauscl  13088  bdbm1.3ii  13089  bj-axemptylem  13090  bj-nalset  13093