Theorem bdsepnf 15824

Description: Version of ax-bdsep 15820 with one disjoint variable condition removed, the other disjoint variable condition replaced by a nonfreeness hypothesis, and without initial universal quantifier. See also bdsepnfALT 15825. Use bdsep1 15821 when sufficient. (Contributed by BJ, 5-Oct-2019.)

Hypotheses

Ref	Expression
bdsepnf.nf	⊢ Ⅎ𝑏𝜑
bdsepnf.1	⊢ BOUNDED 𝜑

Assertion

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

Distinct variable group:   𝑎,𝑏,𝑥

Allowed substitution hints:   𝜑(𝑥,𝑎,𝑏)

Proof of Theorem bdsepnf

Step	Hyp	Ref	Expression
1		bdsepnf.1	. . 3 ⊢ BOUNDED 𝜑
2	1	bdsepnft 15823	. 2 ⊢ (∀𝑥Ⅎ𝑏𝜑 → ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)))
3		bdsepnf.nf	. 2 ⊢ Ⅎ𝑏𝜑
4	2, 3	mpg 1474	1 ⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))

Syntax hints:   ∧ wa 104   ↔ wb 105  ∀wal 1371  Ⅎwnf 1483  ∃wex 1515  BOUNDED wbd 15748

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-bdsep 15820

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-cleq 2198  df-clel 2201

This theorem is referenced by: (None)