Theorem spsbc 2924

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 1749 and rspsbc 2995. (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 1749	. . . 4 |- ( A. x ph -> [ y / x ] ph )
2		sbsbc 2917	. . . 4 |- ( [ y / x ] ph <-> [. y / x ]. ph )
3	1, 2	sylib 121	. . 3 |- ( A. x ph -> [. y / x ]. ph )
4		dfsbcq 2915	. . 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 2760	1 |- ( A e. V -> ( A. x ph -> [. A / x ]. ph ) )

Syntax hints:   -> wi 4   A.wal 1330   = wceq 1332   e. wcel 1481   [wsb 1736   [.wsbc 2913

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 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-v 2691  df-sbc 2914

This theorem is referenced by:  spsbcd  2925  sbcth  2926  sbcthdv  2927  sbceqal  2968  sbcimdv  2978  csbiebt  3044  csbexga  4064