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Mirrors > Home > ILE Home > Th. List > nfrexdxy | GIF version |
Description: Not-free for restricted existential quantification where 𝑥 and 𝑦 are distinct. See nfrexdya 2414 for a version with 𝑦 and 𝐴 distinct instead. (Contributed by Jim Kingdon, 30-May-2018.) |
Ref | Expression |
---|---|
nfraldxy.2 | ⊢ Ⅎ𝑦𝜑 |
nfraldxy.3 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
nfraldxy.4 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfrexdxy | ⊢ (𝜑 → Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rex 2366 | . 2 ⊢ (∃𝑦 ∈ 𝐴 𝜓 ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) | |
2 | nfraldxy.2 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | nfcv 2229 | . . . . . 6 ⊢ Ⅎ𝑥𝑦 | |
4 | 3 | a1i 9 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝑦) |
5 | nfraldxy.3 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
6 | 4, 5 | nfeld 2245 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴) |
7 | nfraldxy.4 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
8 | 6, 7 | nfand 1506 | . . 3 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓)) |
9 | 2, 8 | nfexd 1692 | . 2 ⊢ (𝜑 → Ⅎ𝑥∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) |
10 | 1, 9 | nfxfrd 1410 | 1 ⊢ (𝜑 → Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 Ⅎwnf 1395 ∃wex 1427 ∈ wcel 1439 Ⅎwnfc 2216 ∃wrex 2361 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-4 1446 ax-17 1465 ax-ial 1473 ax-i5r 1474 ax-ext 2071 |
This theorem depends on definitions: df-bi 116 df-nf 1396 df-cleq 2082 df-clel 2085 df-nfc 2218 df-rex 2366 |
This theorem is referenced by: nfrexdya 2414 nfrexxy 2416 nfunid 3666 strcollnft 12152 |
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