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Theorem vtocl 2625
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 1437 . 2 𝑥𝜓
2 vtocl.1 . 2 𝐴 ∈ V
3 vtocl.2 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
4 vtocl.3 . 2 𝜑
51, 2, 3, 4vtoclf 2624 1 𝜓
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102   = wceq 1259  wcel 1409  Vcvv 2574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-v 2576
This theorem is referenced by:  vtoclb  2628  zfauscl  3904  bnd2  3953  pwex  3959  uniex  4201  ordtriexmid  4274  onsucsssucexmid  4279  regexmid  4287  ordsoexmid  4313  onintexmid  4324  reg3exmid  4331  nnregexmid  4369  acexmidlemv  5537  caovcan  5692  findcard2  6376  findcard2s  6377  bj-uniex  10410  bj-omtrans  10454
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