Theorem spsbc 2966

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 1768 and rspsbc 3037. (Contributed by NM, 16-Jan-2004.)

Assertion

Ref	Expression
spsbc	|- ( A e. V -> ( A. x ph -> [. A / x ]. ph ) )

Proof of Theorem spsbc

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		stdpc4 1768	. . . 4 |- ( A. x ph -> [ y / x ] ph )
2		sbsbc 2959	. . . 4 |- ( [ y / x ] ph <-> [. y / x ]. ph )
3	1, 2	sylib 121	. . 3 |- ( A. x ph -> [. y / x ]. ph )
4		dfsbcq 2957	. . 3 |- ( y = A -> ( [. y / x ]. ph <-> [. A / x ]. ph ) )
5	3, 4	syl5ib 153	. 2 |- ( y = A -> ( A. x ph -> [. A / x ]. ph ) )
6	5	vtocleg 2801	1 |- ( A e. V -> ( A. x ph -> [. A / x ]. ph ) )

Syntax hints:   -> wi 4   A.wal 1346   = wceq 1348   [wsb 1755   e. wcel 2141   [.wsbc 2955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-v 2732  df-sbc 2956

This theorem is referenced by:  spsbcd  2967  sbcth  2968  sbcthdv  2969  sbceqal  3010  sbcimdv  3020  csbiebt  3088  csbexga  4117