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Theorem nfbrd 5103
Description: Deduction version of bound-variable hypothesis builder nfbr 5104. (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 5058 . 2 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2 nfbrd.2 . . . 4 (𝜑𝑥𝐴)
3 nfbrd.4 . . . 4 (𝜑𝑥𝐵)
42, 3nfopd 4812 . . 3 (𝜑𝑥𝐴, 𝐵⟩)
5 nfbrd.3 . . 3 (𝜑𝑥𝑅)
64, 5nfeld 2987 . 2 (𝜑 → Ⅎ𝑥𝐴, 𝐵⟩ ∈ 𝑅)
71, 6nfxfrd 1848 1 (𝜑 → Ⅎ𝑥 𝐴𝑅𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wnf 1778  wcel 2108  wnfc 2959  cop 4565   class class class wbr 5057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5058
This theorem is referenced by:  nfbr  5104
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