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Mirrors > Home > ILE Home > Th. List > nfpr | GIF version |
Description: Bound-variable hypothesis builder for unordered pairs. (Contributed by NM, 14-Nov-1995.) |
Ref | Expression |
---|---|
nfpr.1 | ⊢ Ⅎ𝑥𝐴 |
nfpr.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
nfpr | ⊢ Ⅎ𝑥{𝐴, 𝐵} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfpr2 3541 | . 2 ⊢ {𝐴, 𝐵} = {𝑦 ∣ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵)} | |
2 | nfpr.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | nfeq2 2291 | . . . 4 ⊢ Ⅎ𝑥 𝑦 = 𝐴 |
4 | nfpr.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
5 | 4 | nfeq2 2291 | . . . 4 ⊢ Ⅎ𝑥 𝑦 = 𝐵 |
6 | 3, 5 | nfor 1553 | . . 3 ⊢ Ⅎ𝑥(𝑦 = 𝐴 ∨ 𝑦 = 𝐵) |
7 | 6 | nfab 2284 | . 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵)} |
8 | 1, 7 | nfcxfr 2276 | 1 ⊢ Ⅎ𝑥{𝐴, 𝐵} |
Colors of variables: wff set class |
Syntax hints: ∨ wo 697 = wceq 1331 {cab 2123 Ⅎwnfc 2266 {cpr 3523 |
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 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-un 3070 df-sn 3528 df-pr 3529 |
This theorem is referenced by: nfsn 3578 nfop 3716 |
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