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Mirrors > Home > MPE Home > Th. List > sepex | Structured version Visualization version GIF version |
Description: Convert implication to equivalence within an existence statement using the Separation Scheme (Aussonderung) ax-sep 5301. Similar to Theorem 1.3(ii) of [BellMachover] p. 463. (Contributed by Matthew House, 19-Sep-2025.) |
Ref | Expression |
---|---|
sepex | ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 ∈ 𝑦) → ∃𝑧∀𝑥(𝑥 ∈ 𝑧 ↔ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sepexlem 5304 | . 2 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 ∈ 𝑦) → ∃𝑤∀𝑥(𝑥 ∈ 𝑤 ↔ 𝜑)) | |
2 | biimpr 220 | . . . 4 ⊢ ((𝑥 ∈ 𝑤 ↔ 𝜑) → (𝜑 → 𝑥 ∈ 𝑤)) | |
3 | 2 | alimi 1807 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝑤 ↔ 𝜑) → ∀𝑥(𝜑 → 𝑥 ∈ 𝑤)) |
4 | 3 | eximi 1831 | . 2 ⊢ (∃𝑤∀𝑥(𝑥 ∈ 𝑤 ↔ 𝜑) → ∃𝑤∀𝑥(𝜑 → 𝑥 ∈ 𝑤)) |
5 | sepexlem 5304 | . 2 ⊢ (∃𝑤∀𝑥(𝜑 → 𝑥 ∈ 𝑤) → ∃𝑧∀𝑥(𝑥 ∈ 𝑧 ↔ 𝜑)) | |
6 | 1, 4, 5 | 3syl 18 | 1 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 ∈ 𝑦) → ∃𝑧∀𝑥(𝑥 ∈ 𝑧 ↔ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∀wal 1534 ∃wex 1775 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-sep 5301 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1776 |
This theorem is referenced by: sepexi 5306 |
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