ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sbid GIF version

Theorem sbid 1774
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]𝜑𝜑)

Proof of Theorem sbid
StepHypRef Expression
1 equid 1701 . . 3 𝑥 = 𝑥
2 sbequ12 1771 . . 3 (𝑥 = 𝑥 → (𝜑 ↔ [𝑥 / 𝑥]𝜑))
31, 2ax-mp 5 . 2 (𝜑 ↔ [𝑥 / 𝑥]𝜑)
43bicomi 132 1 ([𝑥 / 𝑥]𝜑𝜑)
Colors of variables: wff set class
Syntax hints:  wb 105  [wsb 1762
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-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530
This theorem depends on definitions:  df-bi 117  df-sb 1763
This theorem is referenced by:  abid  2165  sbceq1a  2972  sbcid  2978
  Copyright terms: Public domain W3C validator