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Theorem spsbcd 2799
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 1674 and rspsbc 2868. (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 2798 . 2 (𝐴𝑉 → (∀𝑥𝜓[𝐴 / 𝑥]𝜓))
41, 2, 3sylc 60 1 (𝜑[𝐴 / 𝑥]𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1257  wcel 1409  [wsbc 2787
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-v 2576  df-sbc 2788
This theorem is referenced by:  ovmpt2dxf  5654
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