Theorem sbcth 2950

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 1429	. 2 |- A. x ph
3		spsbc 2948	. 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 1333   e. wcel 2128   [.wsbc 2937

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 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-v 2714  df-sbc 2938

This theorem is referenced by:  rabrsndc  3628  iota4an  5155