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Theorem nfcsbd 3094
Description: Deduction version of nfcsb 3096. (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 3060 . 2 𝐴 / 𝑦𝐵 = {𝑧[𝐴 / 𝑦]𝑧𝐵}
2 nfv 1528 . . 3 𝑧𝜑
3 nfcsbd.1 . . . 4 𝑦𝜑
4 nfcsbd.2 . . . 4 (𝜑𝑥𝐴)
5 nfcsbd.3 . . . . 5 (𝜑𝑥𝐵)
65nfcrd 2333 . . . 4 (𝜑 → Ⅎ𝑥 𝑧𝐵)
73, 4, 6nfsbcd 2984 . . 3 (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝑧𝐵)
82, 7nfabd 2339 . 2 (𝜑𝑥{𝑧[𝐴 / 𝑦]𝑧𝐵})
91, 8nfcxfrd 2317 1 (𝜑𝑥𝐴 / 𝑦𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wnf 1460  wcel 2148  {cab 2163  wnfc 2306  [wsbc 2964  csb 3059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-sbc 2965  df-csb 3060
This theorem is referenced by:  nfcsb  3096
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