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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 2466 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 2420 | . 2 ⊢ (∃𝑦 ∈ 𝐴 𝜓 ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) | |
2 | sban 1926 | . . . . . 6 ⊢ ([𝑧 / 𝑦](𝑦 ∈ 𝐴 ∧ 𝜓) ↔ ([𝑧 / 𝑦]𝑦 ∈ 𝐴 ∧ [𝑧 / 𝑦]𝜓)) | |
3 | clelsb3 2242 | . . . . . . 7 ⊢ ([𝑧 / 𝑦]𝑦 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) | |
4 | 3 | anbi1i 453 | . . . . . 6 ⊢ (([𝑧 / 𝑦]𝑦 ∈ 𝐴 ∧ [𝑧 / 𝑦]𝜓) ↔ (𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑦]𝜓)) |
5 | 2, 4 | bitri 183 | . . . . 5 ⊢ ([𝑧 / 𝑦](𝑦 ∈ 𝐴 ∧ 𝜓) ↔ (𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑦]𝜓)) |
6 | 5 | exbii 1584 | . . . 4 ⊢ (∃𝑧[𝑧 / 𝑦](𝑦 ∈ 𝐴 ∧ 𝜓) ↔ ∃𝑧(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑦]𝜓)) |
7 | nfv 1508 | . . . . 5 ⊢ Ⅎ𝑧(𝑦 ∈ 𝐴 ∧ 𝜓) | |
8 | 7 | sb8e 1829 | . . . 4 ⊢ (∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜓) ↔ ∃𝑧[𝑧 / 𝑦](𝑦 ∈ 𝐴 ∧ 𝜓)) |
9 | df-rex 2420 | . . . 4 ⊢ (∃𝑧 ∈ 𝐴 [𝑧 / 𝑦]𝜓 ↔ ∃𝑧(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑦]𝜓)) | |
10 | 6, 8, 9 | 3bitr4i 211 | . . 3 ⊢ (∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜓) ↔ ∃𝑧 ∈ 𝐴 [𝑧 / 𝑦]𝜓) |
11 | nfv 1508 | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
12 | nfraldya.3 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
13 | nfraldya.2 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
14 | nfraldya.4 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
15 | 13, 14 | nfsbd 1948 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥[𝑧 / 𝑦]𝜓) |
16 | 11, 12, 15 | nfrexdxy 2466 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∃𝑧 ∈ 𝐴 [𝑧 / 𝑦]𝜓) |
17 | 10, 16 | nfxfrd 1451 | . 2 ⊢ (𝜑 → Ⅎ𝑥∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) |
18 | 1, 17 | nfxfrd 1451 | 1 ⊢ (𝜑 → Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 Ⅎwnf 1436 ∃wex 1468 ∈ wcel 1480 [wsb 1735 Ⅎwnfc 2266 ∃wrex 2415 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-cleq 2130 df-clel 2133 df-nfc 2268 df-rex 2420 |
This theorem is referenced by: nfrexya 2472 |
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