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Theorem spsbc 2948
 Description: Specialization: if a formula is true for all sets, it is true for any class which is a set. Similar to Theorem 6.11 of [Quine] p. 44. See also stdpc4 1755 and rspsbc 3019. (Contributed by NM, 16-Jan-2004.)
Assertion
Ref Expression
spsbc (𝐴𝑉 → (∀𝑥𝜑[𝐴 / 𝑥]𝜑))

Proof of Theorem spsbc
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 stdpc4 1755 . . . 4 (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
2 sbsbc 2941 . . . 4 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
31, 2sylib 121 . . 3 (∀𝑥𝜑[𝑦 / 𝑥]𝜑)
4 dfsbcq 2939 . . 3 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
53, 4syl5ib 153 . 2 (𝑦 = 𝐴 → (∀𝑥𝜑[𝐴 / 𝑥]𝜑))
65vtocleg 2783 1 (𝐴𝑉 → (∀𝑥𝜑[𝐴 / 𝑥]𝜑))
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1333   = wceq 1335  [wsb 1742   ∈ wcel 2128  [wsbc 2937 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-v 2714  df-sbc 2938 This theorem is referenced by:  spsbcd  2949  sbcth  2950  sbcthdv  2951  sbceqal  2992  sbcimdv  3002  csbiebt  3070  csbexga  4092
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