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Theorem nfbrd 4628
Description: Deduction version of bound-variable hypothesis builder nfbr 4629. (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 4584 . 2 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2 nfbrd.2 . . . 4 (𝜑𝑥𝐴)
3 nfbrd.4 . . . 4 (𝜑𝑥𝐵)
42, 3nfopd 4357 . . 3 (𝜑𝑥𝐴, 𝐵⟩)
5 nfbrd.3 . . 3 (𝜑𝑥𝑅)
64, 5nfeld 2759 . 2 (𝜑 → Ⅎ𝑥𝐴, 𝐵⟩ ∈ 𝑅)
71, 6nfxfrd 1772 1 (𝜑 → Ⅎ𝑥 𝐴𝑅𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wnf 1699  wcel 1977  wnfc 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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