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Theorem vtoclga 2619
Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 20-Aug-1995.)
Hypotheses
Ref Expression
vtoclga.1 (𝑥 = 𝐴 → (𝜑𝜓))
vtoclga.2 (𝑥𝐵𝜑)
Assertion
Ref Expression
vtoclga (𝐴𝐵𝜓)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem vtoclga
StepHypRef Expression
1 nfcv 2178 . 2 𝑥𝐴
2 nfv 1421 . 2 𝑥𝜓
3 vtoclga.1 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
4 vtoclga.2 . 2 (𝑥𝐵𝜑)
51, 2, 3, 4vtoclgaf 2618 1 (𝐴𝐵𝜓)
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 2559
This theorem is referenced by:  vtoclri  2628  ssuni  3602  ordtriexmid  4247  onsucsssucexmid  4252  tfis3  4309  fvmpt3  5251  fvmptssdm  5255  fnressn  5349  fressnfv  5350  caovord  5672  caovimo  5694  tfrlem1  5923  freccl  5993  nnacl  6059  nnmcl  6060  nnacom  6063  nnaass  6064  nndi  6065  nnmass  6066  nnmsucr  6067  nnmcom  6068  nnsucsssuc  6071  nntri3or  6072  nnaordi  6081  nnaword  6084  nnmordi  6089  nnaordex  6100  findcard  6345  findcard2  6346  findcard2s  6347  indpi  6438  prarloclem3  6593  uzind4s2  8532  cnref1o  8580  frec2uzrdg  9169  expcl2lemap  9241  climub  9838  climserile  9839  ialginv  9860  ialgcvg  9861  ialgcvga  9864  ialgfx  9865
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