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| Mirrors > Home > MPE Home > Th. List > Mathboxes > expandrexn | Structured version Visualization version GIF version | ||
| Description: Expand a restricted existential quantifier to primitives while contracting a double negation. (Contributed by Rohan Ridenour, 13-Aug-2023.) | 
| Ref | Expression | 
|---|---|
| expandrexn.1 | ⊢ (𝜑 ↔ ¬ 𝜓) | 
| Ref | Expression | 
|---|---|
| expandrexn | ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | expandrexn.1 | . . 3 ⊢ (𝜑 ↔ ¬ 𝜓) | |
| 2 | 1 | rexbii 3093 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 ¬ 𝜓) | 
| 3 | df-rex 3070 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝜓)) | |
| 4 | exanali 1858 | . 2 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝜓) ↔ ¬ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)) | |
| 5 | 2, 3, 4 | 3bitri 297 | 1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1537 ∃wex 1778 ∈ wcel 2107 ∃wrex 3069 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 df-rex 3070 | 
| This theorem is referenced by: expandrex 44316 ismnuprim 44318 | 
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