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Theorem sbc8g 3441
Description: This is the closest we can get to df-sbc 3434 if we start from dfsbcq 3435 (see its comments) and dfsbcq2 3436. (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 3435 . 2 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
2 eleq1 2688 . 2 (𝑦 = 𝐴 → (𝑦 ∈ {𝑥𝜑} ↔ 𝐴 ∈ {𝑥𝜑}))
3 df-clab 2608 . . 3 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
4 equid 1938 . . . 4 𝑦 = 𝑦
5 dfsbcq2 3436 . . . 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 3265 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝐴 ∈ {𝑥𝜑}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  [wsb 1879  wcel 1989  {cab 2607  [wsbc 3433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-12 2046  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-v 3200  df-sbc 3434
This theorem is referenced by:  bnj984  31007  rusbcALT  38466
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