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Theorem spsbcd 3059
 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 Mario Carneiro, 9-Feb-2017.)
Hypotheses
Ref Expression
spsbcd.1 (φA V)
spsbcd.2 (φxψ)
Assertion
Ref Expression
spsbcd (φ → [̣A / xψ)

Proof of Theorem spsbcd
StepHypRef Expression
1 spsbcd.1 . 2 (φA V)
2 spsbcd.2 . 2 (φxψ)
3 spsbc 3058 . 2 (A V → (xψ → [̣A / xψ))
41, 2, 3sylc 56 1 (φ → [̣A / xψ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1540   ∈ wcel 1710  [̣wsbc 3046 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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: (None)
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