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Theorem sbc8g 3429
Description: This is the closest we can get to df-sbc 3422 if we start from dfsbcq 3423 (see its comments) and dfsbcq2 3424. (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 3423 . 2 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
2 eleq1 2686 . 2 (𝑦 = 𝐴 → (𝑦 ∈ {𝑥𝜑} ↔ 𝐴 ∈ {𝑥𝜑}))
3 df-clab 2608 . . 3 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
4 equid 1936 . . . 4 𝑦 = 𝑦
5 dfsbcq2 3424 . . . 4 (𝑦 = 𝑦 → ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑))
64, 5ax-mp 5 . . 3 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
73, 6bitr2i 265 . 2 ([𝑦 / 𝑥]𝜑𝑦 ∈ {𝑥𝜑})
81, 2, 7vtoclbg 3256 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝐴 ∈ {𝑥𝜑}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  [wsb 1877  wcel 1987  {cab 2607  [wsbc 3421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-12 2044  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-v 3191  df-sbc 3422
This theorem is referenced by:  bnj984  30765  rusbcALT  38157
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