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Theorem nfbrd 5188
Description: Deduction version of bound-variable hypothesis builder nfbr 5189. (Contributed by NM, 13-Dec-2005.) (Revised by Mario Carneiro, 14-Oct-2016.)
Hypotheses
Ref Expression
nfbrd.2 (𝜑𝑥𝐴)
nfbrd.3 (𝜑𝑥𝑅)
nfbrd.4 (𝜑𝑥𝐵)
Assertion
Ref Expression
nfbrd (𝜑 → Ⅎ𝑥 𝐴𝑅𝐵)

Proof of Theorem nfbrd
StepHypRef Expression
1 df-br 5143 . 2 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2 nfbrd.2 . . . 4 (𝜑𝑥𝐴)
3 nfbrd.4 . . . 4 (𝜑𝑥𝐵)
42, 3nfopd 4889 . . 3 (𝜑𝑥𝐴, 𝐵⟩)
5 nfbrd.3 . . 3 (𝜑𝑥𝑅)
64, 5nfeld 2916 . 2 (𝜑 → Ⅎ𝑥𝐴, 𝐵⟩ ∈ 𝑅)
71, 6nfxfrd 1853 1 (𝜑 → Ⅎ𝑥 𝐴𝑅𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wnf 1782  wcel 2107  wnfc 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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