Mathbox for Rohan Ridenour |
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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 3246 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 ¬ 𝜓) |
3 | df-rex 3143 | . 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 299 | 1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1534 ∃wex 1779 ∈ wcel 2113 ∃wrex 3138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1780 df-rex 3143 |
This theorem is referenced by: expandrex 40702 ismnuprim 40704 |
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