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Mirrors > Home > ILE Home > Th. List > nfbrd | GIF version |
Description: Deduction version of bound-variable hypothesis builder nfbr 3944. (Contributed by NM, 13-Dec-2005.) (Revised by Mario Carneiro, 14-Oct-2016.) |
Ref | Expression |
---|---|
nfbrd.2 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
nfbrd.3 | ⊢ (𝜑 → Ⅎ𝑥𝑅) |
nfbrd.4 | ⊢ (𝜑 → Ⅎ𝑥𝐵) |
Ref | Expression |
---|---|
nfbrd | ⊢ (𝜑 → Ⅎ𝑥 𝐴𝑅𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 3900 | . 2 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
2 | nfbrd.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
3 | nfbrd.4 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
4 | 2, 3 | nfopd 3692 | . . 3 ⊢ (𝜑 → Ⅎ𝑥〈𝐴, 𝐵〉) |
5 | nfbrd.3 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝑅) | |
6 | 4, 5 | nfeld 2274 | . 2 ⊢ (𝜑 → Ⅎ𝑥〈𝐴, 𝐵〉 ∈ 𝑅) |
7 | 1, 6 | nfxfrd 1436 | 1 ⊢ (𝜑 → Ⅎ𝑥 𝐴𝑅𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 Ⅎwnf 1421 ∈ wcel 1465 Ⅎwnfc 2245 〈cop 3500 class class class wbr 3899 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-un 3045 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 |
This theorem is referenced by: nfbr 3944 |
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