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Theorem a4sbcd 2979
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 1897 and ra4sbc 3044. (Contributed by Mario Carneiro, 9-Feb-2017.)
Hypotheses
Ref Expression
a4sbcd.1  |-  ( ph  ->  A  e.  V )
a4sbcd.2  |-  ( ph  ->  A. x ps )
Assertion
Ref Expression
a4sbcd  |-  ( ph  ->  [. A  /  x ]. ps )

Proof of Theorem a4sbcd
StepHypRef Expression
1 a4sbcd.1 . 2  |-  ( ph  ->  A  e.  V )
2 a4sbcd.2 . 2  |-  ( ph  ->  A. x ps )
3 a4sbc 2978 . 2  |-  ( A  e.  V  ->  ( A. x ps  ->  [. A  /  x ]. ps )
)
41, 2, 3sylc 58 1  |-  ( ph  ->  [. A  /  x ]. ps )
Colors of variables: wff set class
Syntax hints:    -> wi 6   A.wal 1532    e. wcel 1621   [.wsbc 2966
This theorem is referenced by:  ex-natded9.26  20796
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-gen 1536  ax-17 1628  ax-12o 1664  ax-9 1684  ax-4 1692  ax-ext 2239
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-v 2765  df-sbc 2967
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