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Theorem vtocl 2608
Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 30-Aug-1993.)
Hypotheses
Ref Expression
vtocl.1 𝐴 ∈ V
vtocl.2 (𝑥 = 𝐴 → (𝜑𝜓))
vtocl.3 𝜑
Assertion
Ref Expression
vtocl 𝜓
Distinct variable groups:   𝑥,𝐴   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem vtocl
StepHypRef Expression
1 nfv 1421 . 2 𝑥𝜓
2 vtocl.1 . 2 𝐴 ∈ V
3 vtocl.2 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
4 vtocl.3 . 2 𝜑
51, 2, 3, 4vtoclf 2607 1 𝜓
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1243  wcel 1393  Vcvv 2557
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-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-v 2559
This theorem is referenced by:  vtoclb  2611  zfauscl  3877  bnd2  3926  pwex  3932  uniex  4174  ordtriexmid  4247  onsucsssucexmid  4252  regexmid  4260  ordsoexmid  4286  onintexmid  4297  reg3exmid  4304  nnregexmid  4342  acexmidlemv  5510  caovcan  5665  findcard2  6346  findcard2s  6347  bj-uniex  10037  bj-omtrans  10081
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