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| Mirrors > Home > MPE Home > Th. List > cbvrmow | Structured version Visualization version GIF version | ||
| Description: Change the bound variable of a restricted at-most-one quantifier using implicit substitution. Version of cbvrmo 3429 with a disjoint variable condition, which does not require ax-10 2141, ax-13 2377. (Contributed by NM, 16-Jun-2017.) Avoid ax-10 2141, ax-13 2377. (Revised by GG, 23-May-2024.) | 
| Ref | Expression | 
|---|---|
| cbvrmow.1 | ⊢ Ⅎ𝑦𝜑 | 
| cbvrmow.2 | ⊢ Ⅎ𝑥𝜓 | 
| cbvrmow.3 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | 
| Ref | Expression | 
|---|---|
| cbvrmow | ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nfv 1914 | . . . 4 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴 | |
| 2 | cbvrmow.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
| 3 | 1, 2 | nfan 1899 | . . 3 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) | 
| 4 | nfv 1914 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 | |
| 5 | cbvrmow.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
| 6 | 4, 5 | nfan 1899 | . . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓) | 
| 7 | eleq1w 2824 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
| 8 | cbvrmow.3 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
| 9 | 7, 8 | anbi12d 632 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑦 ∈ 𝐴 ∧ 𝜓))) | 
| 10 | 3, 6, 9 | cbvmow 2603 | . 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) | 
| 11 | df-rmo 3380 | . 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
| 12 | df-rmo 3380 | . 2 ⊢ (∃*𝑦 ∈ 𝐴 𝜓 ↔ ∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) | |
| 13 | 10, 11, 12 | 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 ∈ wcel 2108 ∃*wmo 2538 ∃*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-clel 2816 df-rmo 3380 | 
| This theorem is referenced by: cbvreuw 3410 cbvdisj 5120 | 
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