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Theorem vtocl 2740
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 1508 . 2 𝑥𝜓
2 vtocl.1 . 2 𝐴 ∈ V
3 vtocl.2 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
4 vtocl.3 . 2 𝜑
51, 2, 3, 4vtoclf 2739 1 𝜓
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1331  wcel 1480  Vcvv 2686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-v 2688
This theorem is referenced by:  vtoclb  2743  zfauscl  4048  bnd2  4097  uniex  4359  ordtriexmid  4437  onsucsssucexmid  4442  regexmid  4450  ordsoexmid  4477  onintexmid  4487  reg3exmid  4494  nnregexmid  4534  acexmidlemv  5772  caovcan  5935  findcard2  6783  findcard2s  6784  inffiexmid  6800  sup3exmid  8722  bj-uniex  13145  bj-omtrans  13184
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