Theorem spsbc 3057

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 1824 and rspsbc 3129. (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 1824	. . . 4 |- ( A. x ph -> [ y / x ] ph )
2		sbsbc 3049	. . . 4 |- ( [ y / x ] ph <-> [. y / x ]. ph )
3	1, 2	sylib 122	. . 3 |- ( A. x ph -> [. y / x ]. ph )
4		dfsbcq 3047	. . 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 2890	1 |- ( A e. V -> ( A. x ph -> [. A / x ]. ph ) )

Syntax hints:   -> wi 4   A.wal 1396   = wceq 1398   [wsb 1811   e. wcel 2205   [.wsbc 3045

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-v 2817  df-sbc 3046

This theorem is referenced by:  spsbcd  3058  sbcth  3059  sbcthdv  3060  sbceqal  3101  sbcimdv  3111  csbiebt  3181  csbexga  4243