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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 5195. (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 5149 | . 2 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
2 | nfbrd.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
3 | nfbrd.4 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
4 | 2, 3 | nfopd 4895 | . . 3 ⊢ (𝜑 → Ⅎ𝑥〈𝐴, 𝐵〉) |
5 | nfbrd.3 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝑅) | |
6 | 4, 5 | nfeld 2915 | . 2 ⊢ (𝜑 → Ⅎ𝑥〈𝐴, 𝐵〉 ∈ 𝑅) |
7 | 1, 6 | nfxfrd 1851 | 1 ⊢ (𝜑 → Ⅎ𝑥 𝐴𝑅𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 Ⅎwnf 1780 ∈ wcel 2106 Ⅎwnfc 2888 〈cop 4637 class class class wbr 5148 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 |
This theorem is referenced by: nfbr 5195 nfttrcld 9748 |
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