Theorem sbid 1747

Description: An identity theorem for substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by NM, 5-Aug-1993.)

Assertion

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

Proof of Theorem sbid

Step	Hyp	Ref	Expression
1		equid 1677	. . 3 |- x = x
2		sbequ12 1744	. . 3 |- ( x = x -> ( ph <-> [ x / x ] ph ) )
3	1, 2	ax-mp 5	. 2 |- ( ph <-> [ x / x ] ph )
4	3	bicomi 131	1 |- ( [ x / x ] ph <-> ph )

Syntax hints:   <-> wb 104   [wsb 1735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510

This theorem depends on definitions:  df-bi 116  df-sb 1736

This theorem is referenced by:  abid  2127  sbceq1a  2918  sbcid  2924