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Theorem sbcieg 2847
Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005.)
Hypothesis
Ref Expression
sbcieg.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
sbcieg (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜓))
Distinct variable groups:   𝑥,𝐴   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem sbcieg
StepHypRef Expression
1 nfv 1462 . 2 𝑥𝜓
2 sbcieg.1 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
31, 2sbciegf 2846 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1285  wcel 1434  [wsbc 2816
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-sbc 2817
This theorem is referenced by:  sbcie  2849  ralsng  3441  rexsng  3442  ralrnmpt  5341  rexrnmpt  5342  nn1suc  8125  cjth  9871  bezoutlemnewy  10529  bezoutlemstep  10530  bezoutlema  10532  bezoutlemb  10533  prmind2  10646
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