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Theorem spsbc 2835
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 1700 and rspsbc 2905. (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 1700 . . . 4 (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
2 sbsbc 2828 . . . 4 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
31, 2sylib 120 . . 3 (∀𝑥𝜑[𝑦 / 𝑥]𝜑)
4 dfsbcq 2826 . . 3 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
53, 4syl5ib 152 . 2 (𝑦 = 𝐴 → (∀𝑥𝜑[𝐴 / 𝑥]𝜑))
65vtocleg 2678 1 (𝐴𝑉 → (∀𝑥𝜑[𝐴 / 𝑥]𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1283   = wceq 1285  wcel 1434  [wsb 1687  [wsbc 2824
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 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-v 2612  df-sbc 2825
This theorem is referenced by:  spsbcd  2836  sbcth  2837  sbcthdv  2838  sbceqal  2878  sbcimdv  2888  csbiebt  2951  csbexga  3926
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