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

Theorem sbidm 2413
Description: An idempotent law for substitution. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 13-Jul-2019.)
Assertion
Ref Expression
sbidm ([𝑦 / 𝑥][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Proof of Theorem sbidm
StepHypRef Expression
1 sbcom3 2410 . 2 ([𝑦 / 𝑥][𝑥 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑥]𝜑)
2 sbid 2111 . . 3 ([𝑥 / 𝑥]𝜑𝜑)
32sbbii 1884 . 2 ([𝑦 / 𝑥][𝑥 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
41, 3bitr3i 266 1 ([𝑦 / 𝑥][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 196  [wsb 1877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-12 2044  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1702  df-nf 1707  df-sb 1878
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator