Theorem bdsep1 17077

Description: Version of ax-bdsep 17076 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 17076	. 2 ⊢ ∀𝑎∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))
3	2	spi 1589	1 ⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))

This proof depends on syntax axioms:   ∧ wa 104   ↔ wb 105  ∀wal 1400  ∃wex 1545  BOUNDED wbd 17004

This proof depends on axioms:  ax-mp 5  ax-4 1563  ax-bdsep 17076

This theorem is used by:  bdsep2  17078  bdsepg  17082  bdbm1.3ii  17083  bj-axemptylem  17084  bj-nalset  17087