MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbid Structured version   Visualization version   GIF version

Theorem sbid 2254
Description: 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.)
Assertion
Ref Expression
sbid ([𝑥 / 𝑥]𝜑𝜑)

Proof of Theorem sbid
StepHypRef Expression
1 equid 2010 . 2 𝑥 = 𝑥
2 sbequ12r 2251 . 2 (𝑥 = 𝑥 → ([𝑥 / 𝑥]𝜑𝜑))
31, 2ax-mp 5 1 ([𝑥 / 𝑥]𝜑𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 206  [wsb 2063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-12 2176
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-sb 2064
This theorem is referenced by:  sbcovOLD  2256  sbco  2511  sbidm  2514  abid  2717  sbceq1a  3798  sbcid  3804  frege58bid  43920  ichid  47443  sbidd  49292  sbidd-misc  49293
  Copyright terms: Public domain W3C validator