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Theorem spsbcd 3805
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 2066 and rspsbc 3888. (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 3804 . 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 1535  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:  ovmpodxf  7583  ex-natded9.26  30448  spsbcdi  38105  ovmpordxf  48184
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