Theorem sbid

An identity theorem for substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 30-Sep-2018.)

Conclusion
\( \vdash [ x / x ] \, \varphi \leftrightarrow \varphi \)
Proof
l0
equid
\( \vdash x = x \)
sbequ12r
\( \vdash x = x \rightarrow ([ x / x ] \, \varphi \leftrightarrow \varphi) \)
ax-mp l0 _