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Mirrors > Home > MPE Home > Th. List > sepexi | Structured version Visualization version GIF version |
Description: Convert implication to equivalence within an existence statement using the Separation Scheme (Aussonderung) ax-sep 5301. Inference associated with sepex 5305. (Contributed by NM, 21-Jun-1993.) Generalize conclusion, extract closed form, and avoid ax-9 2115. (Revised by Matthew House, 19-Sep-2025.) |
Ref | Expression |
---|---|
sepexi.1 | ⊢ ∃𝑦∀𝑥(𝜑 → 𝑥 ∈ 𝑦) |
Ref | Expression |
---|---|
sepexi | ⊢ ∃𝑧∀𝑥(𝑥 ∈ 𝑧 ↔ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sepexi.1 | . 2 ⊢ ∃𝑦∀𝑥(𝜑 → 𝑥 ∈ 𝑦) | |
2 | sepex 5305 | . 2 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 ∈ 𝑦) → ∃𝑧∀𝑥(𝑥 ∈ 𝑧 ↔ 𝜑)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ∃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: axpow3 5373 vpwex 5382 axpr 5432 zfpair2 5438 axun2 7755 uniex2 7756 |
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