Theorem sbid 1184

Description: An identity theorem for substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint).

Assertion

Ref	Expression
sbid	|- ([x / x]ph <-> ph)

Proof of Theorem sbid

Step	Hyp	Ref	Expression
1		equid 1126	. . 3 |- x = x
2		sbequ12 1181	. . 3 |- (x = x -> (ph <-> [x / x]ph))
3	1, 2	ax-mp 7	. 2 |- (ph <-> [x / x]ph)
4	3	bicomi 172	1 |- ([x / x]ph <-> ph)

Syntax hints:   <-> wb 146  [wsbc 1170

This theorem is referenced by:  abid 1465  sbceq1a 1944  csbid 2005

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-12 968  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172

Copyright terms: Public domain