Theorem spsbcd 3010

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 1797 and rspsbc 3080. (Contributed by Mario Carneiro, 9-Feb-2017.)

Hypotheses

Ref	Expression
spsbcd.1	|- ( ph -> A e. V )
spsbcd.2	|- ( ph -> A. x ps )

Assertion

Ref	Expression
spsbcd	|- ( ph -> [. A / x ]. ps )

Proof of Theorem spsbcd

Step	Hyp	Ref	Expression
1		spsbcd.1	. 2 |- ( ph -> A e. V )
2		spsbcd.2	. 2 |- ( ph -> A. x ps )
3		spsbc 3009	. 2 |- ( A e. V -> ( A. x ps -> [. A / x ]. ps ) )
4	1, 2, 3	sylc 62	1 |- ( ph -> [. A / x ]. ps )

Syntax hints:   -> wi 4   A.wal 1370   e. wcel 2175   [.wsbc 2997

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-v 2773  df-sbc 2998

This theorem is referenced by:  ovmpodxf  6070