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Theorem sbciegf 2870
Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
Hypotheses
Ref Expression
sbciegf.1 𝑥𝜓
sbciegf.2 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
sbciegf (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜓))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem sbciegf
StepHypRef Expression
1 sbciegf.1 . 2 𝑥𝜓
2 sbciegf.2 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
32ax-gen 1383 . 2 𝑥(𝑥 = 𝐴 → (𝜑𝜓))
4 sbciegft 2869 . 2 ((𝐴𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → ([𝐴 / 𝑥]𝜑𝜓))
51, 3, 4mp3an23 1265 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1287   = wceq 1289  wnf 1394  wcel 1438  [wsbc 2840
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-sbc 2841
This theorem is referenced by:  sbcieg  2871  iunxsngf  3805  opelopabf  4099  eqerlem  6313
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