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| Mirrors > Home > MPE Home > Th. List > nfrexdw | Structured version Visualization version GIF version | ||
| Description: Deduction version of nfrexw 3312. (Contributed by Mario Carneiro, 14-Oct-2016.) Add disjoint variable condition to avoid ax-13 2376. See nfrexd 3372 for a less restrictive version requiring more axioms. (Revised by GG, 20-Jan-2024.) | 
| Ref | Expression | 
|---|---|
| nfraldw.1 | ⊢ Ⅎ𝑦𝜑 | 
| nfraldw.2 | ⊢ (𝜑 → Ⅎ𝑥𝐴) | 
| nfraldw.3 | ⊢ (𝜑 → Ⅎ𝑥𝜓) | 
| Ref | Expression | 
|---|---|
| nfrexdw | ⊢ (𝜑 → Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜓) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | dfrex2 3072 | . 2 ⊢ (∃𝑦 ∈ 𝐴 𝜓 ↔ ¬ ∀𝑦 ∈ 𝐴 ¬ 𝜓) | |
| 2 | nfraldw.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
| 3 | nfraldw.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
| 4 | nfraldw.3 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
| 5 | 4 | nfnd 1857 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝜓) | 
| 6 | 2, 3, 5 | nfraldw 3308 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 ¬ 𝜓) | 
| 7 | 6 | nfnd 1857 | . 2 ⊢ (𝜑 → Ⅎ𝑥 ¬ ∀𝑦 ∈ 𝐴 ¬ 𝜓) | 
| 8 | 1, 7 | nfxfrd 1853 | 1 ⊢ (𝜑 → Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜓) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 Ⅎwnf 1782 Ⅎwnfc 2889 ∀wral 3060 ∃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 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-10 2140 ax-11 2156 ax-12 2176 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ex 1779 df-nf 1783 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 | 
| This theorem is referenced by: nfrexw 3312 nfunid 4912 nfttrcld 9751 nfiund 49248 | 
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