Theorem sbt 1830

Description: A substitution into a theorem remains true. (See chvar 1803 and chvarv 1988 for versions using implicit substitition.) (Contributed by NM, 21-Jan-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypothesis

Ref	Expression
sbt.1	|- ph

Assertion

Ref	Expression
sbt	|- [ y / x ] ph

Proof of Theorem sbt

Step	Hyp	Ref	Expression
1		sbt.1	. 2 |- ph
2	1	nfth 1510	. . 3 |- F/ x ph
3	2	sbf 1823	. 2 |- ( [ y / x ] ph <-> ph )
4	1, 3	mpbir 146	1 |- [ y / x ] ph

Syntax hints:   [wsb 1808

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-i9 1576  ax-ial 1580

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809

This theorem is referenced by:  vjust  2800