Theorem sbcth 2968

Description: A substitution into a theorem remains true (when A is a set). (Contributed by NM, 5-Nov-2005.)

Hypothesis

Ref	Expression
sbcth.1	|- ph

Assertion

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

Proof of Theorem sbcth

Step	Hyp	Ref	Expression
1		sbcth.1	. . 3 |- ph
2	1	ax-gen 1442	. 2 |- A. x ph
3		spsbc 2966	. 2 |- ( A e. V -> ( A. x ph -> [. A / x ]. ph ) )
4	2, 3	mpi 15	1 |- ( A e. V -> [. A / x ]. ph )

Syntax hints:   -> wi 4   A.wal 1346   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:  rabrsndc  3651  iota4an  5179