Theorem spsbcd 3064

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 1828 and rspsbc 3135. (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 3063	. 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 1400   e. wcel 2209   [.wsbc 3051

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 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-v 2823  df-sbc 3052

This theorem is referenced by:  ovmpodxf  6204