Theorem spsbcd 3018

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 1799 and rspsbc 3089. (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 3017	. 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 1371   e. wcel 2178   [.wsbc 3005

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 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-v 2778  df-sbc 3006

This theorem is referenced by:  ovmpodxf  6094