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Theorem sbc8g 2885
Description: This is the closest we can get to df-sbc 2879 if we start from dfsbcq 2880 (see its comments) and dfsbcq2 2881. (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 2880 . 2 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
2 eleq1 2177 . 2 (𝑦 = 𝐴 → (𝑦 ∈ {𝑥𝜑} ↔ 𝐴 ∈ {𝑥𝜑}))
3 df-clab 2102 . . 3 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
4 equid 1660 . . . 4 𝑦 = 𝑦
5 dfsbcq2 2881 . . . 4 (𝑦 = 𝑦 → ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑))
64, 5ax-mp 7 . . 3 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
73, 6bitr2i 184 . 2 ([𝑦 / 𝑥]𝜑𝑦 ∈ {𝑥𝜑})
81, 2, 7vtoclbg 2718 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝐴 ∈ {𝑥𝜑}))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wcel 1463  [wsb 1718  {cab 2101  [wsbc 2878
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-sbc 2879
This theorem is referenced by:  bj-elssuniab  12688
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