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Theorem vtoclg 2630
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 2194 . 2 𝑥𝐴
2 nfv 1437 . 2 𝑥𝜓
3 vtoclg.1 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
4 vtoclg.2 . 2 𝜑
51, 2, 3, 4vtoclgf 2629 1 (𝐴𝑉𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102   = wceq 1259  wcel 1409
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-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576
This theorem is referenced by:  vtoclbg  2631  ceqex  2694  mo2icl  2743  nelrdva  2769  sbctt  2852  sbcnestgf  2925  csbing  3172  prnzg  3520  sneqrg  3561  unisng  3625  csbopabg  3863  trss  3891  inex1g  3921  ssexg  3924  pwexg  3961  prexgOLD  3974  prexg  3975  opth  4002  ordelord  4146  uniexg  4203  vtoclr  4416  resieq  4650  csbima12g  4714  dmsnsnsng  4826  iota5  4915  csbiotag  4923  funmo  4945  fconstg  5111  funfveu  5216  funbrfv  5240  fnbrfvb  5242  fvelimab  5257  ssimaexg  5263  fvelrn  5326  isoselem  5487  csbriotag  5508  csbov123g  5571  ovg  5667  tfrexlem  5979  rdg0g  6006  ensn1g  6308  fundmeng  6318  xpdom2g  6337  phplem3g  6350  prcdnql  6640  prcunqu  6641  prdisj  6648  shftvalg  9665  shftval4g  9666  climshft  10056  bdzfauscl  10397  bdinex1g  10408  bdssexg  10411  bj-prexg  10418  bj-uniexg  10425
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