Theorem sbcth 2988

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 1459	. 2 |- A. x ph
3		spsbc 2986	. 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 1361   e. wcel 2158   [.wsbc 2974

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 1457  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-v 2751  df-sbc 2975

This theorem is referenced by:  rabrsndc  3672  iota4an  5209