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Mirrors > Home > ILE Home > Th. List > nfiotadw | GIF version |
Description: Bound-variable hypothesis builder for the ℩ class. (Contributed by Jim Kingdon, 21-Dec-2018.) |
Ref | Expression |
---|---|
nfiotadw.1 | ⊢ Ⅎ𝑦𝜑 |
nfiotadw.2 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfiotadw | ⊢ (𝜑 → Ⅎ𝑥(℩𝑦𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfiota2 5089 | . 2 ⊢ (℩𝑦𝜓) = ∪ {𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)} | |
2 | nfv 1508 | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
3 | nfiotadw.1 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
4 | nfiotadw.2 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
5 | nfcv 2281 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑦 | |
6 | nfcv 2281 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑧 | |
7 | 5, 6 | nfeq 2289 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑦 = 𝑧 |
8 | 7 | a1i 9 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑥 𝑦 = 𝑧) |
9 | 4, 8 | nfbid 1567 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥(𝜓 ↔ 𝑦 = 𝑧)) |
10 | 3, 9 | nfald 1733 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝜓 ↔ 𝑦 = 𝑧)) |
11 | 2, 10 | nfabd 2300 | . . 3 ⊢ (𝜑 → Ⅎ𝑥{𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)}) |
12 | 11 | nfunid 3743 | . 2 ⊢ (𝜑 → Ⅎ𝑥∪ {𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)}) |
13 | 1, 12 | nfcxfrd 2279 | 1 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1329 = wceq 1331 Ⅎwnf 1436 {cab 2125 Ⅎwnfc 2268 ∪ cuni 3736 ℩cio 5086 |
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 |
This theorem is referenced by: nfiotaw 5092 nfriotadxy 5738 |
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