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| Mirrors > Home > MPE Home > Th. List > nfrmow | Structured version Visualization version GIF version | ||
| Description: Bound-variable hypothesis builder for restricted uniqueness. Version of nfrmo 3434 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by NM, 16-Jun-2017.) Avoid ax-13 2377. (Revised by GG, 10-Jan-2024.) Avoid ax-9 2118, ax-ext 2708. (Revised by Wolf Lammen, 21-Nov-2024.) | 
| Ref | Expression | 
|---|---|
| nfrmow.1 | ⊢ Ⅎ𝑥𝐴 | 
| nfrmow.2 | ⊢ Ⅎ𝑥𝜑 | 
| Ref | Expression | 
|---|---|
| nfrmow | ⊢ Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜑 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-rmo 3380 | . 2 ⊢ (∃*𝑦 ∈ 𝐴 𝜑 ↔ ∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)) | |
| 2 | nfrmow.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
| 3 | 2 | nfcri 2897 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 | 
| 4 | nfrmow.2 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
| 5 | 3, 4 | nfan 1899 | . . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑) | 
| 6 | 5 | nfmov 2560 | . 2 ⊢ Ⅎ𝑥∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜑) | 
| 7 | 1, 6 | nfxfr 1853 | 1 ⊢ Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜑 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 Ⅎwnf 1783 ∈ wcel 2108 ∃*wmo 2538 Ⅎwnfc 2890 ∃*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-10 2141 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-nfc 2892 df-rmo 3380 | 
| This theorem is referenced by: 2rmorex 3760 2reurex 3766 | 
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