Theorem sbcth 2964

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 1437	. 2 |- A. x ph
3		spsbc 2962	. 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 1341   e. wcel 2136   [.wsbc 2951

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 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-v 2728  df-sbc 2952

This theorem is referenced by:  rabrsndc  3644  iota4an  5172