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Theorem sbc8g 2845
Description: This is the closest we can get to df-sbc 2839 if we start from dfsbcq 2840 (see its comments) and dfsbcq2 2841. (Contributed by NM, 18-Nov-2008.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.)
Assertion
Ref Expression
sbc8g (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝐴 ∈ {𝑥𝜑}))

Proof of Theorem sbc8g
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfsbcq 2840 . 2 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
2 eleq1 2150 . 2 (𝑦 = 𝐴 → (𝑦 ∈ {𝑥𝜑} ↔ 𝐴 ∈ {𝑥𝜑}))
3 df-clab 2075 . . 3 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
4 equid 1634 . . . 4 𝑦 = 𝑦
5 dfsbcq2 2841 . . . 4 (𝑦 = 𝑦 → ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑))
64, 5ax-mp 7 . . 3 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
73, 6bitr2i 183 . 2 ([𝑦 / 𝑥]𝜑𝑦 ∈ {𝑥𝜑})
81, 2, 7vtoclbg 2680 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝐴 ∈ {𝑥𝜑}))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wcel 1438  [wsb 1692  {cab 2074  [wsbc 2838
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-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 2839
This theorem is referenced by:  bj-elssuniab  11337
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