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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 5180 | . 2 ⊢ (℩𝑦𝜓) = ∪ {𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)} | |
2 | nfv 1528 | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
3 | nfiotadw.1 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
4 | nfiotadw.2 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
5 | nfcv 2319 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑦 | |
6 | nfcv 2319 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑧 | |
7 | 5, 6 | nfeq 2327 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑦 = 𝑧 |
8 | 7 | a1i 9 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑥 𝑦 = 𝑧) |
9 | 4, 8 | nfbid 1588 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥(𝜓 ↔ 𝑦 = 𝑧)) |
10 | 3, 9 | nfald 1760 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝜓 ↔ 𝑦 = 𝑧)) |
11 | 2, 10 | nfabd 2339 | . . 3 ⊢ (𝜑 → Ⅎ𝑥{𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)}) |
12 | 11 | nfunid 3817 | . 2 ⊢ (𝜑 → Ⅎ𝑥∪ {𝑧 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑧)}) |
13 | 1, 12 | nfcxfrd 2317 | 1 ⊢ (𝜑 → Ⅎ𝑥(℩𝑦𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∀wal 1351 = wceq 1353 Ⅎwnf 1460 {cab 2163 Ⅎwnfc 2306 ∪ cuni 3810 ℩cio 5177 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rex 2461 df-sn 3599 df-uni 3811 df-iota 5179 |
This theorem is referenced by: nfiotaw 5183 nfriotadxy 5839 |
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