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Theorem nfcsbd 3169
 Description: Deduction version of nfcsb 3170. (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 (φx[A / y]B)

Proof of Theorem nfcsbd
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 df-csb 3137 . 2 [A / y]B = {z A / yz B}
2 nfv 1619 . . 3 zφ
3 nfcsbd.1 . . . 4 yφ
4 nfcsbd.2 . . . 4 (φxA)
5 nfcsbd.3 . . . . 5 (φxB)
65nfcrd 2502 . . . 4 (φ → Ⅎx z B)
73, 4, 6nfsbcd 3066 . . 3 (φ → ℲxA / yz B)
82, 7nfabd 2508 . 2 (φx{z A / yz B})
91, 8nfcxfrd 2487 1 (φx[A / y]B)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  Ⅎwnf 1544   ∈ wcel 1710  {cab 2339  Ⅎwnfc 2476  [̣wsbc 3046  [csb 3136 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-sbc 3047  df-csb 3137 This theorem is referenced by:  nfcsb  3170
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