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Theorem nfbrd 3835
Description: Deduction version of bound-variable hypothesis builder nfbr 3836. (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 3793 . 2 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2 nfbrd.2 . . . 4 (𝜑𝑥𝐴)
3 nfbrd.4 . . . 4 (𝜑𝑥𝐵)
42, 3nfopd 3594 . . 3 (𝜑𝑥𝐴, 𝐵⟩)
5 nfbrd.3 . . 3 (𝜑𝑥𝑅)
64, 5nfeld 2209 . 2 (𝜑 → Ⅎ𝑥𝐴, 𝐵⟩ ∈ 𝑅)
71, 6nfxfrd 1380 1 (𝜑 → Ⅎ𝑥 𝐴𝑅𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wnf 1365  wcel 1409  wnfc 2181  cop 3406   class class class wbr 3792
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793
This theorem is referenced by:  nfbr  3836
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