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Theorem vtocl 2791
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 1528 . 2 𝑥𝜓
2 vtocl.1 . 2 𝐴 ∈ V
3 vtocl.2 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
4 vtocl.3 . 2 𝜑
51, 2, 3, 4vtoclf 2790 1 𝜓
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1353  wcel 2148  Vcvv 2737
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-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  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-v 2739
This theorem is referenced by:  vtoclb  2794  zfauscl  4120  bnd2  4170  uniex  4433  ordtriexmid  4516  onsucsssucexmid  4522  regexmid  4530  ordsoexmid  4557  onintexmid  4568  reg3exmid  4575  nnregexmid  4616  acexmidlemv  5866  caovcan  6032  findcard2  6882  findcard2s  6883  inffiexmid  6899  sup3exmid  8890  bj-uniex  14291  bj-omtrans  14330
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