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| Mirrors > Home > MPE Home > Th. List > cbvreuw | Structured version Visualization version GIF version | ||
| Description: Change the bound variable of a restricted unique existential quantifier using implicit substitution. Version of cbvreu 3428 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by Mario Carneiro, 15-Oct-2016.) Avoid ax-13 2377. (Revised by GG, 10-Jan-2024.) Avoid ax-10 2141. (Revised by Wolf Lammen, 10-Dec-2024.) |
| Ref | Expression |
|---|---|
| cbvreuw.1 | ⊢ Ⅎ𝑦𝜑 |
| cbvreuw.2 | ⊢ Ⅎ𝑥𝜓 |
| cbvreuw.3 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
| Ref | Expression |
|---|---|
| cbvreuw | ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cbvreuw.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
| 2 | cbvreuw.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
| 3 | cbvreuw.3 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
| 4 | 1, 2, 3 | cbvrexw 3307 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓) |
| 5 | 1, 2, 3 | cbvrmow 3409 | . . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓) |
| 6 | 4, 5 | anbi12i 628 | . 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐴 𝜑) ↔ (∃𝑦 ∈ 𝐴 𝜓 ∧ ∃*𝑦 ∈ 𝐴 𝜓)) |
| 7 | reu5 3382 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐴 𝜑)) | |
| 8 | reu5 3382 | . 2 ⊢ (∃!𝑦 ∈ 𝐴 𝜓 ↔ (∃𝑦 ∈ 𝐴 𝜓 ∧ ∃*𝑦 ∈ 𝐴 𝜓)) | |
| 9 | 6, 7, 8 | 3bitr4i 303 | 1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 Ⅎwnf 1783 ∃wrex 3070 ∃!wreu 3378 ∃*wrmo 3379 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-11 2157 ax-12 2177 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-mo 2540 df-eu 2569 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 |
| This theorem is referenced by: reu8nf 3877 poimirlem25 37652 |
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