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Theorem vtoclga 2636
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 2194 . 2 𝑥𝐴
2 nfv 1437 . 2 𝑥𝜓
3 vtoclga.1 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
4 vtoclga.2 . 2 (𝑥𝐵𝜑)
51, 2, 3, 4vtoclgaf 2635 1 (𝐴𝐵𝜓)
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:  vtoclri  2645  ssuni  3630  ordtriexmid  4275  onsucsssucexmid  4280  tfis3  4337  fvmpt3  5279  fvmptssdm  5283  fnressn  5377  fressnfv  5378  caovord  5700  caovimo  5722  tfrlem1  5954  freccl  6024  nnacl  6090  nnmcl  6091  nnacom  6094  nnaass  6095  nndi  6096  nnmass  6097  nnmsucr  6098  nnmcom  6099  nnsucsssuc  6102  nntri3or  6103  nnaordi  6112  nnaword  6115  nnmordi  6120  nnaordex  6131  findcard  6376  findcard2  6377  findcard2s  6378  indpi  6498  prarloclem3  6653  uzind4s2  8630  cnref1o  8680  frec2uzrdg  9359  expcl2lemap  9432  climub  10095  climserile  10096  ialginv  10269  ialgcvg  10270  ialgcvga  10273  ialgfx  10274
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