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Theorem spsbc 3058
 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 2024 and rspsbc 3124. (Contributed by NM, 16-Jan-2004.)
Assertion
Ref Expression
spsbc (A V → (xφ → [̣A / xφ))

Proof of Theorem spsbc
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 stdpc4 2024 . . . 4 (xφ → [y / x]φ)
2 sbsbc 3050 . . . 4 ([y / x]φ ↔ [̣y / xφ)
31, 2sylib 188 . . 3 (xφ → [̣y / xφ)
4 dfsbcq 3048 . . 3 (y = A → ([̣y / xφ ↔ [̣A / xφ))
53, 4syl5ib 210 . 2 (y = A → (xφ → [̣A / xφ))
65vtocleg 2925 1 (A V → (xφ → [̣A / xφ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1540   = wceq 1642  [wsb 1648   ∈ wcel 1710  [̣wsbc 3046 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2861  df-sbc 3047 This theorem is referenced by:  spsbcd  3059  sbcth  3060  sbcthdv  3061  sbceqal  3097  sbcimdv  3107  csbexg  3146  csbiebt  3172
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