Theorem bdsepnf 10008

Description: Version of ax-bdsep 10004 with one DV condition removed, the other DV condition replaced by a non-freeness hypothesis, and without initial universal quantifier. See also bdsepnfALT 10009. Use bdsep1 10005 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 10007	. 2 ⊢ (∀𝑥Ⅎ𝑏𝜑 → ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)))
3		bdsepnf.nf	. 2 ⊢ Ⅎ𝑏𝜑
4	2, 3	mpg 1340	1 ⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))

Syntax hints:   ∧ wa 97   ↔ wb 98  ∀wal 1241  Ⅎwnf 1349  ∃wex 1381  BOUNDED wbd 9932

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-bdsep 10004

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-cleq 2033  df-clel 2036

This theorem is referenced by: (None)