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Mirrors > Home > ILE Home > Th. List > nfriotadxy | GIF version |
Description: Deduction version of nfriota 5655. (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 5646 | . 2 ⊢ (℩𝑦 ∈ 𝐴 𝜓) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) | |
2 | nfriotadxy.1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | nfcv 2235 | . . . . . 6 ⊢ Ⅎ𝑥𝑦 | |
4 | 3 | a1i 9 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝑦) |
5 | nfriotadxy.3 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
6 | 4, 5 | nfeld 2251 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴) |
7 | nfriotadxy.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
8 | 6, 7 | nfand 1512 | . . 3 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓)) |
9 | 2, 8 | nfiotadxy 5017 | . 2 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))) |
10 | 1, 9 | nfcxfrd 2233 | 1 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦 ∈ 𝐴 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 Ⅎwnf 1401 ∈ wcel 1445 Ⅎwnfc 2222 ℩cio 5012 ℩crio 5645 |
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-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-rex 2376 df-sn 3472 df-uni 3676 df-iota 5014 df-riota 5646 |
This theorem is referenced by: nfriota 5655 |
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