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Theorem spsbcd 3789
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 2064 and rspsbc 3871. (Contributed by Mario Carneiro, 9-Feb-2017.)
Hypotheses
Ref Expression
spsbcd.1 (𝜑𝐴𝑉)
spsbcd.2 (𝜑 → ∀𝑥𝜓)
Assertion
Ref Expression
spsbcd (𝜑[𝐴 / 𝑥]𝜓)

Proof of Theorem spsbcd
StepHypRef Expression
1 spsbcd.1 . 2 (𝜑𝐴𝑉)
2 spsbcd.2 . 2 (𝜑 → ∀𝑥𝜓)
3 spsbc 3788 . 2 (𝐴𝑉 → (∀𝑥𝜓[𝐴 / 𝑥]𝜓))
41, 2, 3sylc 65 1 (𝜑[𝐴 / 𝑥]𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1532  wcel 2099  [wsbc 3775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-sbc 3776
This theorem is referenced by:  ovmpodxf  7568  ex-natded9.26  30349  spsbcdi  37832  ovmpordxf  47753
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