ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sbequ12r GIF version

Theorem sbequ12r 1730
Description: An equality theorem for substitution. (Contributed by NM, 6-Oct-2004.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
Assertion
Ref Expression
sbequ12r (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑𝜑))

Proof of Theorem sbequ12r
StepHypRef Expression
1 sbequ12 1729 . . 3 (𝑦 = 𝑥 → (𝜑 ↔ [𝑥 / 𝑦]𝜑))
21bicomd 140 . 2 (𝑦 = 𝑥 → ([𝑥 / 𝑦]𝜑𝜑))
32equcoms 1669 1 (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  [wsb 1720
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 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-4 1472  ax-17 1491  ax-i9 1495
This theorem depends on definitions:  df-bi 116  df-sb 1721
This theorem is referenced by:  abbi  2231  findes  4487  opeliunxp  4564  isarep1  5179  bezoutlemmain  11613
  Copyright terms: Public domain W3C validator