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Theorem spsbc 2769
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 1655 and rspsbc 2834. (Contributed by NM, 16-Jan-2004.)
Assertion
Ref Expression
spsbc (A 𝑉 → (xφ[A / x]φ))

Proof of Theorem spsbc
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 stdpc4 1655 . . . 4 (xφ → [y / x]φ)
2 sbsbc 2762 . . . 4 ([y / x]φ[y / x]φ)
31, 2sylib 127 . . 3 (xφ[y / x]φ)
4 dfsbcq 2760 . . 3 (y = A → ([y / x]φ[A / x]φ))
53, 4syl5ib 143 . 2 (y = A → (xφ[A / x]φ))
65vtocleg 2618 1 (A 𝑉 → (xφ[A / x]φ))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1240   = wceq 1242   wcel 1390  [wsb 1642  [wsbc 2758
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553  df-sbc 2759
This theorem is referenced by:  spsbcd  2770  sbcth  2771  sbcthdv  2772  sbceqal  2808  sbcimdv  2817  csbiebt  2880  csbexga  3876
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