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Theorem sbciegf 3009
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 1460 . 2 𝑥(𝑥 = 𝐴 → (𝜑𝜓))
4 sbciegft 3008 . 2 ((𝐴𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → ([𝐴 / 𝑥]𝜑𝜓))
51, 3, 4mp3an23 1340 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wal 1362   = wceq 1364  wnf 1471  wcel 2160  [wsbc 2977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-sbc 2978
This theorem is referenced by:  sbcieg  3010  iunxsngf  3979  opelopabf  4289  eqerlem  6584
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