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Theorem spsbcd 2963
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 1763 and rspsbc 3033. (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 2962 . 2 (𝐴𝑉 → (∀𝑥𝜓[𝐴 / 𝑥]𝜓))
41, 2, 3sylc 62 1 (𝜑[𝐴 / 𝑥]𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1341  wcel 2136  [wsbc 2951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-ext 2147
This theorem depends on definitions:  df-bi 116  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-v 2728  df-sbc 2952
This theorem is referenced by:  ovmpodxf  5967
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