Theorem sepex 5255

Description: Convert implication to equivalence within an existence statement using the Separation Scheme (Aussonderung) ax-sep 5249. Similar to Theorem 1.3(ii) of [BellMachover] p. 463. (Contributed by Matthew House, 19-Sep-2025.)

Assertion

Ref	Expression
sepex	⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 ∈ 𝑦) → ∃𝑧∀𝑥(𝑥 ∈ 𝑧 ↔ 𝜑))

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

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem sepex

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sepexlem 5254	. 2 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 ∈ 𝑦) → ∃𝑤∀𝑥(𝑥 ∈ 𝑤 ↔ 𝜑))
2		biimpr 223	. . . 4 ⊢ ((𝑥 ∈ 𝑤 ↔ 𝜑) → (𝜑 → 𝑥 ∈ 𝑤))
3	2	alimi 1844	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝑤 ↔ 𝜑) → ∀𝑥(𝜑 → 𝑥 ∈ 𝑤))
4	3	eximi 1868	. 2 ⊢ (∃𝑤∀𝑥(𝑥 ∈ 𝑤 ↔ 𝜑) → ∃𝑤∀𝑥(𝜑 → 𝑥 ∈ 𝑤))
5		sepexlem 5254	. 2 ⊢ (∃𝑤∀𝑥(𝜑 → 𝑥 ∈ 𝑤) → ∃𝑧∀𝑥(𝑥 ∈ 𝑧 ↔ 𝜑))
6	1, 4, 5	3syl 19	1 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 ∈ 𝑦) → ∃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:  sepexi  5256