Theorem sepexi 5256

Description: Convert implication to equivalence within an existence statement using the Separation Scheme (Aussonderung) ax-sep 5249. Inference associated with sepex 5255. (Contributed by NM, 21-Jun-1993.) Generalize conclusion, extract closed form, and avoid ax-9 2155. (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 5255	. 2 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 ∈ 𝑦) → ∃𝑧∀𝑥(𝑥 ∈ 𝑧 ↔ 𝜑))
3	1, 2	ax-mp 5	1 ⊢ ∃𝑧∀𝑥(𝑥 ∈ 𝑧 ↔ 𝜑)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-sep 5249

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813

This theorem is used by:  axpow3  5330  vpwex  5339  axpr  5389  zfpair2  5392  prex  5396  axun2  7742  uniex2OLD  7744  elirrvOLD  9576