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Mirrors > Home > MPE Home > Th. List > nfrexd | Structured version Visualization version GIF version |
Description: Deduction version of nfrex 3228. (Contributed by Mario Carneiro, 14-Oct-2016.) Add disjoint variable condition to avoid ax-13 2371. See nfrexdg 3227 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.) |
Ref | Expression |
---|---|
nfrexd.1 | ⊢ Ⅎ𝑦𝜑 |
nfrexd.2 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
nfrexd.3 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfrexd | ⊢ (𝜑 → Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfrex2 3161 | . 2 ⊢ (∃𝑦 ∈ 𝐴 𝜓 ↔ ¬ ∀𝑦 ∈ 𝐴 ¬ 𝜓) | |
2 | nfrexd.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
3 | nfrexd.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
4 | nfrexd.3 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
5 | 4 | nfnd 1866 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝜓) |
6 | 2, 3, 5 | nfraldw 3144 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 ¬ 𝜓) |
7 | 6 | nfnd 1866 | . 2 ⊢ (𝜑 → Ⅎ𝑥 ¬ ∀𝑦 ∈ 𝐴 ¬ 𝜓) |
8 | 1, 7 | nfxfrd 1861 | 1 ⊢ (𝜑 → Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 Ⅎwnf 1791 Ⅎwnfc 2884 ∀wral 3061 ∃wrex 3062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-10 2141 ax-11 2158 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-ex 1788 df-nf 1792 df-clel 2816 df-nfc 2886 df-ral 3066 df-rex 3067 |
This theorem is referenced by: nfrex 3228 nfunid 4825 nfttrcld 33509 nfiund 46051 |
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