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Mirrors > Home > MPE Home > Th. List > nfbrd | Structured version Visualization version GIF version |
Description: Deduction version of bound-variable hypothesis builder nfbr 4629. (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 4584 | . 2 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
2 | nfbrd.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
3 | nfbrd.4 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
4 | 2, 3 | nfopd 4357 | . . 3 ⊢ (𝜑 → Ⅎ𝑥〈𝐴, 𝐵〉) |
5 | nfbrd.3 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝑅) | |
6 | 4, 5 | nfeld 2759 | . 2 ⊢ (𝜑 → Ⅎ𝑥〈𝐴, 𝐵〉 ∈ 𝑅) |
7 | 1, 6 | nfxfrd 1772 | 1 ⊢ (𝜑 → Ⅎ𝑥 𝐴𝑅𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 Ⅎwnf 1699 ∈ wcel 1977 Ⅎwnfc 2738 〈cop 4131 class class class wbr 4583 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 |
This theorem is referenced by: nfbr 4629 |
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