Theorem sbcth 3061

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 1546	. 2 |- A.xph
3		spsbc 3059	. 2 |- (A e. V -> (A.xph -> [.A / x ].ph))
4	2, 3	mpi 16	1 |- (A e. V -> [.A / x ].ph)

Syntax hints:   -> wi 4  A.wal 1540   e. wcel 1710   [.wsbc 3047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2862  df-sbc 3048

This theorem is referenced by:  iota4an  4359