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

Theorem sbidm 2548
 Description: An idempotent law for substitution. Usage of this theorem is discouraged because it depends on ax-13 2386. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 13-Jul-2019.) (New usage is discouraged.)
Assertion
Ref Expression
sbidm ([𝑦 / 𝑥][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Proof of Theorem sbidm
StepHypRef Expression
1 sbcom3 2544 . 2 ([𝑦 / 𝑥][𝑥 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑥]𝜑)
2 sbid 2253 . . 3 ([𝑥 / 𝑥]𝜑𝜑)
32sbbii 2077 . 2 ([𝑦 / 𝑥][𝑥 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
41, 3bitr3i 279 1 ([𝑦 / 𝑥][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 208  [wsb 2065 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-10 2141  ax-12 2173  ax-13 2386 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1777  df-nf 1781  df-sb 2066 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator