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Theorem spsbc 2889
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 1731 and rspsbc 2959. (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 1731 . . . 4 (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
2 sbsbc 2882 . . . 4 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
31, 2sylib 121 . . 3 (∀𝑥𝜑[𝑦 / 𝑥]𝜑)
4 dfsbcq 2880 . . 3 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
53, 4syl5ib 153 . 2 (𝑦 = 𝐴 → (∀𝑥𝜑[𝐴 / 𝑥]𝜑))
65vtocleg 2728 1 (𝐴𝑉 → (∀𝑥𝜑[𝐴 / 𝑥]𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1312   = wceq 1314  wcel 1463  [wsb 1718  [wsbc 2878
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-v 2659  df-sbc 2879
This theorem is referenced by:  spsbcd  2890  sbcth  2891  sbcthdv  2892  sbceqal  2932  sbcimdv  2942  csbiebt  3005  csbexga  4016
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