Theorem spsbc 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 3090. (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 1799	. . . 4 |- ( A. x ph -> [ y / x ] ph )
2		sbsbc 3010	. . . 4 |- ( [ y / x ] ph <-> [. y / x ]. ph )
3	1, 2	sylib 122	. . 3 |- ( A. x ph -> [. y / x ]. ph )
4		dfsbcq 3008	. . 3 |- ( y = A -> ( [. y / x ]. ph <-> [. A / x ]. ph ) )
5	3, 4	imbitrid 154	. 2 |- ( y = A -> ( A. x ph -> [. A / x ]. ph ) )
6	5	vtocleg 2852	1 |- ( A e. V -> ( A. x ph -> [. A / x ]. ph ) )

Syntax hints:   -> wi 4   A.wal 1371   = wceq 1373   [wsb 1786   e. wcel 2178   [.wsbc 3006

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 2779  df-sbc 3007

This theorem is referenced by:  spsbcd  3019  sbcth  3020  sbcthdv  3021  sbceqal  3062  sbcimdv  3072  csbiebt  3142  csbexga  4189