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Mirrors > Home > ILE Home > Th. List > nfrexdya | GIF version |
Description: Not-free for restricted existential quantification where 𝑦 and 𝐴 are distinct. See nfrexdxy 2357 for a version with 𝑥 and 𝑦 distinct instead. (Contributed by Jim Kingdon, 30-May-2018.) |
Ref | Expression |
---|---|
nfraldya.2 | ⊢ Ⅎ𝑦𝜑 |
nfraldya.3 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
nfraldya.4 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfrexdya | ⊢ (𝜑 → Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rex 2312 | . 2 ⊢ (∃𝑦 ∈ 𝐴 𝜓 ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) | |
2 | sban 1829 | . . . . . 6 ⊢ ([𝑧 / 𝑦](𝑦 ∈ 𝐴 ∧ 𝜓) ↔ ([𝑧 / 𝑦]𝑦 ∈ 𝐴 ∧ [𝑧 / 𝑦]𝜓)) | |
3 | clelsb3 2142 | . . . . . . 7 ⊢ ([𝑧 / 𝑦]𝑦 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) | |
4 | 3 | anbi1i 431 | . . . . . 6 ⊢ (([𝑧 / 𝑦]𝑦 ∈ 𝐴 ∧ [𝑧 / 𝑦]𝜓) ↔ (𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑦]𝜓)) |
5 | 2, 4 | bitri 173 | . . . . 5 ⊢ ([𝑧 / 𝑦](𝑦 ∈ 𝐴 ∧ 𝜓) ↔ (𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑦]𝜓)) |
6 | 5 | exbii 1496 | . . . 4 ⊢ (∃𝑧[𝑧 / 𝑦](𝑦 ∈ 𝐴 ∧ 𝜓) ↔ ∃𝑧(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑦]𝜓)) |
7 | nfv 1421 | . . . . 5 ⊢ Ⅎ𝑧(𝑦 ∈ 𝐴 ∧ 𝜓) | |
8 | 7 | sb8e 1737 | . . . 4 ⊢ (∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜓) ↔ ∃𝑧[𝑧 / 𝑦](𝑦 ∈ 𝐴 ∧ 𝜓)) |
9 | df-rex 2312 | . . . 4 ⊢ (∃𝑧 ∈ 𝐴 [𝑧 / 𝑦]𝜓 ↔ ∃𝑧(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑦]𝜓)) | |
10 | 6, 8, 9 | 3bitr4i 201 | . . 3 ⊢ (∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜓) ↔ ∃𝑧 ∈ 𝐴 [𝑧 / 𝑦]𝜓) |
11 | nfv 1421 | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
12 | nfraldya.3 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
13 | nfraldya.2 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
14 | nfraldya.4 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
15 | 13, 14 | nfsbd 1851 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥[𝑧 / 𝑦]𝜓) |
16 | 11, 12, 15 | nfrexdxy 2357 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∃𝑧 ∈ 𝐴 [𝑧 / 𝑦]𝜓) |
17 | 10, 16 | nfxfrd 1364 | . 2 ⊢ (𝜑 → Ⅎ𝑥∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) |
18 | 1, 17 | nfxfrd 1364 | 1 ⊢ (𝜑 → Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 Ⅎwnf 1349 ∃wex 1381 ∈ wcel 1393 [wsb 1645 Ⅎwnfc 2165 ∃wrex 2307 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rex 2312 |
This theorem is referenced by: nfrexya 2363 |
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