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Theorem nfbrd 3973
Description: Deduction version of bound-variable hypothesis builder nfbr 3974. (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 3930 . 2 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2 nfbrd.2 . . . 4 (𝜑𝑥𝐴)
3 nfbrd.4 . . . 4 (𝜑𝑥𝐵)
42, 3nfopd 3722 . . 3 (𝜑𝑥𝐴, 𝐵⟩)
5 nfbrd.3 . . 3 (𝜑𝑥𝑅)
64, 5nfeld 2297 . 2 (𝜑 → Ⅎ𝑥𝐴, 𝐵⟩ ∈ 𝑅)
71, 6nfxfrd 1451 1 (𝜑 → Ⅎ𝑥 𝐴𝑅𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wnf 1436  wcel 1480  wnfc 2268  cop 3530   class class class wbr 3929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930
This theorem is referenced by:  nfbr  3974
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