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Mirrors > Home > MPE Home > Th. List > sepexlem | Structured version Visualization version GIF version |
Description: Lemma for sepex 5305. Use sepex 5305 instead. (Contributed by Matthew House, 19-Sep-2025.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sepexlem | ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 ∈ 𝑦) → ∃𝑧∀𝑥(𝑥 ∈ 𝑧 ↔ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-sep 5301 | . . 3 ⊢ ∃𝑧∀𝑥(𝑥 ∈ 𝑧 ↔ (𝑥 ∈ 𝑦 ∧ 𝜑)) | |
2 | bimsc1 844 | . . . . . 6 ⊢ (((𝜑 → 𝑥 ∈ 𝑦) ∧ (𝑥 ∈ 𝑧 ↔ (𝑥 ∈ 𝑦 ∧ 𝜑))) → (𝑥 ∈ 𝑧 ↔ 𝜑)) | |
3 | 2 | ex 412 | . . . . 5 ⊢ ((𝜑 → 𝑥 ∈ 𝑦) → ((𝑥 ∈ 𝑧 ↔ (𝑥 ∈ 𝑦 ∧ 𝜑)) → (𝑥 ∈ 𝑧 ↔ 𝜑))) |
4 | 3 | al2imi 1811 | . . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 ∈ 𝑦) → (∀𝑥(𝑥 ∈ 𝑧 ↔ (𝑥 ∈ 𝑦 ∧ 𝜑)) → ∀𝑥(𝑥 ∈ 𝑧 ↔ 𝜑))) |
5 | 4 | eximdv 1914 | . . 3 ⊢ (∀𝑥(𝜑 → 𝑥 ∈ 𝑦) → (∃𝑧∀𝑥(𝑥 ∈ 𝑧 ↔ (𝑥 ∈ 𝑦 ∧ 𝜑)) → ∃𝑧∀𝑥(𝑥 ∈ 𝑧 ↔ 𝜑))) |
6 | 1, 5 | mpi 20 | . 2 ⊢ (∀𝑥(𝜑 → 𝑥 ∈ 𝑦) → ∃𝑧∀𝑥(𝑥 ∈ 𝑧 ↔ 𝜑)) |
7 | 6 | exlimiv 1927 | 1 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 ∈ 𝑦) → ∃𝑧∀𝑥(𝑥 ∈ 𝑧 ↔ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀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: sepex 5305 |
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