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Theorem sbc8g 3410
 Description: This is the closest we can get to df-sbc 3403 if we start from dfsbcq 3404 (see its comments) and dfsbcq2 3405. (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 3404 . 2 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
2 eleq1 2676 . 2 (𝑦 = 𝐴 → (𝑦 ∈ {𝑥𝜑} ↔ 𝐴 ∈ {𝑥𝜑}))
3 df-clab 2597 . . 3 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
4 equid 1926 . . . 4 𝑦 = 𝑦
5 dfsbcq2 3405 . . . 4 (𝑦 = 𝑦 → ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑))
64, 5ax-mp 5 . . 3 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
73, 6bitr2i 264 . 2 ([𝑦 / 𝑥]𝜑𝑦 ∈ {𝑥𝜑})
81, 2, 7vtoclbg 3240 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝐴 ∈ {𝑥𝜑}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  [wsb 1867   ∈ wcel 1977  {cab 2596  [wsbc 3402 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175  df-sbc 3403 This theorem is referenced by:  bnj984  30276  rusbcALT  37662
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