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Theorem nfsbcd 2777
Description: Deduction version of nfsbc 2778. (Contributed by NM, 23-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.)
Hypotheses
Ref Expression
nfsbcd.1 yφ
nfsbcd.2 (φxA)
nfsbcd.3 (φ → Ⅎxψ)
Assertion
Ref Expression
nfsbcd (φ → Ⅎx[A / y]ψ)

Proof of Theorem nfsbcd
StepHypRef Expression
1 df-sbc 2759 . 2 ([A / y]ψA {yψ})
2 nfsbcd.2 . . 3 (φxA)
3 nfsbcd.1 . . . 4 yφ
4 nfsbcd.3 . . . 4 (φ → Ⅎxψ)
53, 4nfabd 2193 . . 3 (φx{yψ})
62, 5nfeld 2190 . 2 (φ → Ⅎx A {yψ})
71, 6nfxfrd 1361 1 (φ → Ⅎx[A / y]ψ)
Colors of variables: wff set class
Syntax hints:  wi 4  wnf 1346   wcel 1390  {cab 2023  wnfc 2162  [wsbc 2758
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-sbc 2759
This theorem is referenced by:  nfsbc  2778  nfcsbd  2877  sbcnestgf  2891
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