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Mirrors > Home > ILE Home > Th. List > nfunid | GIF version |
Description: Deduction version of nfuni 3708. (Contributed by NM, 18-Feb-2013.) |
Ref | Expression |
---|---|
nfunid.3 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
Ref | Expression |
---|---|
nfunid | ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfuni2 3704 | . 2 ⊢ ∪ 𝐴 = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧} | |
2 | nfv 1491 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | nfv 1491 | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
4 | nfunid.3 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
5 | nfvd 1492 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝑧) | |
6 | 3, 4, 5 | nfrexdxy 2442 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧) |
7 | 2, 6 | nfabd 2274 | . 2 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧}) |
8 | 1, 7 | nfcxfrd 2253 | 1 ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 {cab 2101 Ⅎwnfc 2242 ∃wrex 2391 ∪ cuni 3702 |
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 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 |
This theorem depends on definitions: df-bi 116 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-rex 2396 df-uni 3703 |
This theorem is referenced by: dfnfc2 3720 nfiotadxy 5049 |
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