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Theorem vtoclg 2610
Description: Implicit substitution of a class expression for a setvar variable. (Contributed by NM, 17-Apr-1995.)
Hypotheses
Ref Expression
vtoclg.1 (𝑥 = 𝐴 → (𝜑𝜓))
vtoclg.2 𝜑
Assertion
Ref Expression
vtoclg (𝐴𝑉𝜓)
Distinct variable groups:   𝑥,𝐴   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem vtoclg
StepHypRef Expression
1 nfcv 2178 . 2 𝑥𝐴
2 nfv 1421 . 2 𝑥𝜓
3 vtoclg.1 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
4 vtoclg.2 . 2 𝜑
51, 2, 3, 4vtoclgf 2609 1 (𝐴𝑉𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1243  wcel 1393
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-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2556
This theorem is referenced by:  vtoclbg  2611  ceqex  2668  mo2icl  2717  nelrdva  2743  sbctt  2821  sbcnestgf  2894  csbing  3141  prnzg  3489  sneqrg  3530  unisng  3594  csbopabg  3832  trss  3860  inex1g  3890  ssexg  3893  pwexg  3930  prexgOLD  3943  prexg  3944  opth  3971  ordelord  4105  uniexg  4162  vtoclr  4366  resieq  4600  csbima12g  4664  dmsnsnsng  4776  iota5  4865  csbiotag  4873  funmo  4895  fconstg  5061  funfveu  5166  funbrfv  5190  fnbrfvb  5192  fvelimab  5207  ssimaexg  5213  fvelrn  5276  isoselem  5437  csbriotag  5458  csbov123g  5521  ovg  5617  tfrexlem  5926  rdg0g  5953  ensn1g  6255  fundmeng  6265  xpdom2g  6284  phplem3g  6297  prcdnql  6554  prcunqu  6555  prdisj  6562  shftvalg  9306  shftval4g  9307  climshft  9693  bdzfauscl  9874  bdinex1g  9885  bdssexg  9888  bj-prexg  9895  bj-uniexg  9902
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