Theorem sbcth 2777

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 1338	. 2 |- A. x ph
3		spsbc 2775	. 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 1241   e. wcel 1393   [.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:  rabrsndc  3438  iota4an  4886