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

Theorem sbcid 3439
 Description: An identity theorem for substitution. See sbid 2111. (Contributed by Mario Carneiro, 18-Feb-2017.)
Assertion
Ref Expression
sbcid ([𝑥 / 𝑥]𝜑𝜑)

Proof of Theorem sbcid
StepHypRef Expression
1 sbsbc 3426 . 2 ([𝑥 / 𝑥]𝜑[𝑥 / 𝑥]𝜑)
2 sbid 2111 . 2 ([𝑥 / 𝑥]𝜑𝜑)
31, 2bitr3i 266 1 ([𝑥 / 𝑥]𝜑𝜑)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196  [wsb 1877  [wsbc 3422 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-9 1996  ax-12 2044  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-sbc 3423 This theorem is referenced by:  csbid  3527  snfil  21608  ex-natded9.26  27164  bnj605  30738  dedths  33767  frege93  37771
 Copyright terms: Public domain W3C validator