Theorem sepexi 5250

Description: Convert implication to equivalence within an existence statement using the Separation Scheme (Aussonderung) ax-sep 5245. Inference associated with sepex 5249. (Contributed by NM, 21-Jun-1993.) Generalize conclusion, extract closed form, and avoid ax-9 2151. (Revised by Matthew House, 19-Sep-2025.)

Hypothesis

Ref	Expression
sepexi.1	⊢ ∃𝑦∀𝑥(𝜑 → 𝑥 ∈ 𝑦)

Assertion

Ref	Expression
sepexi	⊢ ∃𝑧∀𝑥(𝑥 ∈ 𝑧 ↔ 𝜑)

Distinct variable groups:   𝑥,𝑦   𝑥,𝑧   𝜑,𝑦   𝜑,𝑧

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem sepexi

Step	Hyp	Ref	Expression
1		sepexi.1	. 2 ⊢ ∃𝑦∀𝑥(𝜑 → 𝑥 ∈ 𝑦)
2		sepex 5249	. 2 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 ∈ 𝑦) → ∃𝑧∀𝑥(𝑥 ∈ 𝑧 ↔ 𝜑))
3	1, 2	ax-mp 5	1 ⊢ ∃𝑧∀𝑥(𝑥 ∈ 𝑧 ↔ 𝜑)

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1557  ∃wex 1798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-sep 5245

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1799

This theorem is referenced by:  axpow3  5324  vpwex  5333  axpr  5383  zfpair2  5390  prex  5394  axun2  7716  uniex2  7717  elirrvOLD  9543