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Theorem spsbc 3804
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. This is Frege's ninth axiom per Proposition 58 of [Frege1879] p. 51. See also stdpc4 2066 and rspsbc 3888. (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 2066 . . . 4 (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
2 sbsbc 3795 . . . 4 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
31, 2sylib 218 . . 3 (∀𝑥𝜑[𝑦 / 𝑥]𝜑)
4 dfsbcq 3793 . . 3 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
53, 4imbitrid 244 . 2 (𝑦 = 𝐴 → (∀𝑥𝜑[𝐴 / 𝑥]𝜑))
65vtocleg 3553 1 (𝐴𝑉 → (∀𝑥𝜑[𝐴 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1535   = wceq 1537  [wsb 2062  wcel 2106  [wsbc 3791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-sbc 3792
This theorem is referenced by:  spsbcd  3805  sbcth  3806  sbcthdv  3807  sbceqalOLD  3858  sbcimdvOLD  3866  csbiebt  3938  csbexg  5316  pm14.18  44424  sbcbi  44537  onfrALTlem3  44542  sbc3orgVD  44849  sbcbiVD  44874  csbingVD  44882  onfrALTlem3VD  44885  csbeq2gVD  44890  csbunigVD  44896
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