Theorem spsbc 2775

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 1658 and rspsbc 2840. (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 1658	. . . 4 |- ( A. x ph -> [ y / x ] ph )
2		sbsbc 2768	. . . 4 |- ( [ y / x ] ph <-> [. y / x ]. ph )
3	1, 2	sylib 127	. . 3 |- ( A. x ph -> [. y / x ]. ph )
4		dfsbcq 2766	. . 3 |- ( y = A -> ( [. y / x ]. ph <-> [. A / x ]. ph ) )
5	3, 4	syl5ib 143	. 2 |- ( y = A -> ( A. x ph -> [. A / x ]. ph ) )
6	5	vtocleg 2624	1 |- ( A e. V -> ( A. x ph -> [. A / x ]. ph ) )

Syntax hints:   -> wi 4   A.wal 1241   = wceq 1243   e. wcel 1393   [wsb 1645   [.wsbc 2764

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-v 2559  df-sbc 2765

This theorem is referenced by:  spsbcd  2776  sbcth  2777  sbcthdv  2778  sbceqal  2814  sbcimdv  2823  csbiebt  2886  csbexga  3885