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| Description: Deduction version of bound-variable hypothesis builder nfbr 5189. (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 5143 | . 2 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
| 2 | nfbrd.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
| 3 | nfbrd.4 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
| 4 | 2, 3 | nfopd 4889 | . . 3 ⊢ (𝜑 → Ⅎ𝑥〈𝐴, 𝐵〉) | 
| 5 | nfbrd.3 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝑅) | |
| 6 | 4, 5 | nfeld 2916 | . 2 ⊢ (𝜑 → Ⅎ𝑥〈𝐴, 𝐵〉 ∈ 𝑅) | 
| 7 | 1, 6 | nfxfrd 1853 | 1 ⊢ (𝜑 → Ⅎ𝑥 𝐴𝑅𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 Ⅎwnf 1782 ∈ wcel 2107 Ⅎwnfc 2889 〈cop 4631 class class class wbr 5142 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 | 
| This theorem is referenced by: nfbr 5189 nfttrcld 9751 | 
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