Theorem sepexlem 5254

Description: Lemma for sepex 5255. Use sepex 5255 instead. (Contributed by Matthew House, 19-Sep-2025.) (New usage is discouraged.)

Assertion

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

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

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem sepexlem

Step	Hyp	Ref	Expression
1		ax-sep 5251	. . 3 ⊢ ∃𝑧∀𝑥(𝑥 ∈ 𝑧 ↔ (𝑥 ∈ 𝑦 ∧ 𝜑))
2		bimsc1 857	. . . . . 6 ⊢ (((𝜑 → 𝑥 ∈ 𝑦) ∧ (𝑥 ∈ 𝑧 ↔ (𝑥 ∈ 𝑦 ∧ 𝜑))) → (𝑥 ∈ 𝑧 ↔ 𝜑))
3	2	ex 417	. . . . 5 ⊢ ((𝜑 → 𝑥 ∈ 𝑦) → ((𝑥 ∈ 𝑧 ↔ (𝑥 ∈ 𝑦 ∧ 𝜑)) → (𝑥 ∈ 𝑧 ↔ 𝜑)))
4	3	al2imi 1838	. . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 ∈ 𝑦) → (∀𝑥(𝑥 ∈ 𝑧 ↔ (𝑥 ∈ 𝑦 ∧ 𝜑)) → ∀𝑥(𝑥 ∈ 𝑧 ↔ 𝜑)))
5	4	eximdv 1940	. . 3 ⊢ (∀𝑥(𝜑 → 𝑥 ∈ 𝑦) → (∃𝑧∀𝑥(𝑥 ∈ 𝑧 ↔ (𝑥 ∈ 𝑦 ∧ 𝜑)) → ∃𝑧∀𝑥(𝑥 ∈ 𝑧 ↔ 𝜑)))
6	1, 5	mpi 21	. 2 ⊢ (∀𝑥(𝜑 → 𝑥 ∈ 𝑦) → ∃𝑧∀𝑥(𝑥 ∈ 𝑧 ↔ 𝜑))
7	6	exlimiv 1953	1 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 ∈ 𝑦) → ∃𝑧∀𝑥(𝑥 ∈ 𝑧 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1561  ∃wex 1802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-sep 5251

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1803

This theorem is referenced by:  sepex  5255