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Mirrors > Home > ILE Home > Th. List > nfriotadxy | GIF version |
Description: Deduction version of nfriota 5739. (Contributed by Jim Kingdon, 12-Jan-2019.) |
Ref | Expression |
---|---|
nfriotadxy.1 | ⊢ Ⅎ𝑦𝜑 |
nfriotadxy.2 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
nfriotadxy.3 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
Ref | Expression |
---|---|
nfriotadxy | ⊢ (𝜑 → Ⅎ𝑥(℩𝑦 ∈ 𝐴 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-riota 5730 | . 2 ⊢ (℩𝑦 ∈ 𝐴 𝜓) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) | |
2 | nfriotadxy.1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | nfcv 2281 | . . . . . 6 ⊢ Ⅎ𝑥𝑦 | |
4 | 3 | a1i 9 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝑦) |
5 | nfriotadxy.3 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
6 | 4, 5 | nfeld 2297 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴) |
7 | nfriotadxy.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
8 | 6, 7 | nfand 1547 | . . 3 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓)) |
9 | 2, 8 | nfiotadw 5091 | . 2 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))) |
10 | 1, 9 | nfcxfrd 2279 | 1 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦 ∈ 𝐴 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 Ⅎwnf 1436 ∈ wcel 1480 Ⅎwnfc 2268 ℩cio 5086 ℩crio 5729 |
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 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-rex 2422 df-sn 3533 df-uni 3737 df-iota 5088 df-riota 5730 |
This theorem is referenced by: nfriota 5739 |
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