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Mirrors > Home > MPE Home > Th. List > nfrexdw | Structured version Visualization version GIF version |
Description: Deduction version of nfrexw 3304. (Contributed by Mario Carneiro, 14-Oct-2016.) Add disjoint variable condition to avoid ax-13 2365. See nfrexd 3363 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 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 3067 | . 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 1853 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝜓) |
6 | 2, 3, 5 | nfraldw 3300 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 ¬ 𝜓) |
7 | 6 | nfnd 1853 | . 2 ⊢ (𝜑 → Ⅎ𝑥 ¬ ∀𝑦 ∈ 𝐴 ¬ 𝜓) |
8 | 1, 7 | nfxfrd 1848 | 1 ⊢ (𝜑 → Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 Ⅎwnf 1777 Ⅎwnfc 2877 ∀wral 3055 ∃wrex 3064 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-10 2129 ax-11 2146 ax-12 2163 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-ex 1774 df-nf 1778 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 |
This theorem is referenced by: nfrexw 3304 nfunid 4908 nfttrcld 9704 nfiund 47974 |
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