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Theorem spsbc 2851
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 1705 and rspsbc 2921. (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 1705 . . . 4 (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
2 sbsbc 2844 . . . 4 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
31, 2sylib 120 . . 3 (∀𝑥𝜑[𝑦 / 𝑥]𝜑)
4 dfsbcq 2842 . . 3 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
53, 4syl5ib 152 . 2 (𝑦 = 𝐴 → (∀𝑥𝜑[𝐴 / 𝑥]𝜑))
65vtocleg 2690 1 (𝐴𝑉 → (∀𝑥𝜑[𝐴 / 𝑥]𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1287   = wceq 1289  wcel 1438  [wsb 1692  [wsbc 2840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-v 2621  df-sbc 2841
This theorem is referenced by:  spsbcd  2852  sbcth  2853  sbcthdv  2854  sbceqal  2894  sbcimdv  2904  csbiebt  2967  csbexga  3967
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