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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 5077. (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 5031 | . 2 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
2 | nfbrd.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
3 | nfbrd.4 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
4 | 2, 3 | nfopd 4782 | . . 3 ⊢ (𝜑 → Ⅎ𝑥〈𝐴, 𝐵〉) |
5 | nfbrd.3 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝑅) | |
6 | 4, 5 | nfeld 2966 | . 2 ⊢ (𝜑 → Ⅎ𝑥〈𝐴, 𝐵〉 ∈ 𝑅) |
7 | 1, 6 | nfxfrd 1855 | 1 ⊢ (𝜑 → Ⅎ𝑥 𝐴𝑅𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 Ⅎwnf 1785 ∈ wcel 2111 Ⅎwnfc 2936 〈cop 4531 class class class wbr 5030 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 |
This theorem is referenced by: nfbr 5077 |
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