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Mirrors > Home > ILE Home > Th. List > nfbrd | GIF version |
Description: Deduction version of bound-variable hypothesis builder nfbr 3969. (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 3925 | . 2 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
2 | nfbrd.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
3 | nfbrd.4 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
4 | 2, 3 | nfopd 3717 | . . 3 ⊢ (𝜑 → Ⅎ𝑥〈𝐴, 𝐵〉) |
5 | nfbrd.3 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝑅) | |
6 | 4, 5 | nfeld 2295 | . 2 ⊢ (𝜑 → Ⅎ𝑥〈𝐴, 𝐵〉 ∈ 𝑅) |
7 | 1, 6 | nfxfrd 1451 | 1 ⊢ (𝜑 → Ⅎ𝑥 𝐴𝑅𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 Ⅎwnf 1436 ∈ wcel 1480 Ⅎwnfc 2266 〈cop 3525 class class class wbr 3924 |
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-3an 964 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 df-op 3531 df-br 3925 |
This theorem is referenced by: nfbr 3969 |
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