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Theorem nfcsbd 2877
Description: Deduction version of nfcsb 2878. (Contributed by NM, 21-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.)
Hypotheses
Ref Expression
nfcsbd.1 yφ
nfcsbd.2 (φxA)
nfcsbd.3 (φxB)
Assertion
Ref Expression
nfcsbd (φxA / yB)

Proof of Theorem nfcsbd
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 df-csb 2847 . 2 A / yB = {z[A / y]z B}
2 nfv 1418 . . 3 zφ
3 nfcsbd.1 . . . 4 yφ
4 nfcsbd.2 . . . 4 (φxA)
5 nfcsbd.3 . . . . 5 (φxB)
65nfcrd 2188 . . . 4 (φ → Ⅎx z B)
73, 4, 6nfsbcd 2777 . . 3 (φ → Ⅎx[A / y]z B)
82, 7nfabd 2193 . 2 (φx{z[A / y]z B})
91, 8nfcxfrd 2173 1 (φxA / yB)
Colors of variables: wff set class
Syntax hints:  wi 4  wnf 1346   wcel 1390  {cab 2023  wnfc 2162  [wsbc 2758  csb 2846
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  df-csb 2847
This theorem is referenced by:  nfcsb  2878
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