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Theorem nfbrd 3777
 Description: Deduction version of bound-variable hypothesis builder nfbr 3778. (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 3735 . 2 (A𝑅B ↔ ⟨A, B 𝑅)
2 nfbrd.2 . . . 4 (φxA)
3 nfbrd.4 . . . 4 (φxB)
42, 3nfopd 3536 . . 3 (φxA, B⟩)
5 nfbrd.3 . . 3 (φx𝑅)
64, 5nfeld 2171 . 2 (φ → ℲxA, B 𝑅)
71, 6nfxfrd 1340 1 (φ → Ⅎx A𝑅B)
 Colors of variables: wff set class Syntax hints:   → wi 4  Ⅎwnf 1325   ∈ wcel 1370  Ⅎwnfc 2143  ⟨cop 3349   class class class wbr 3734 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000 This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-un 2895  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735 This theorem is referenced by:  nfbr  3778
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