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

Theorem sbcid 3723
Description: An identity theorem for substitution. See sbid 2220. (Contributed by Mario Carneiro, 18-Feb-2017.)
Assertion
Ref Expression
sbcid ([𝑥 / 𝑥]𝜑𝜑)

Proof of Theorem sbcid
StepHypRef Expression
1 sbsbc 3710 . 2 ([𝑥 / 𝑥]𝜑[𝑥 / 𝑥]𝜑)
2 sbid 2220 . 2 ([𝑥 / 𝑥]𝜑𝜑)
31, 2bitr3i 278 1 ([𝑥 / 𝑥]𝜑𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 207  [wsb 2042  [wsbc 3706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-12 2141  ax-ext 2769
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1762  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-sbc 3707
This theorem is referenced by:  csbid  3823  snfil  22156  ex-natded9.26  27890  bnj605  31795  dedths  35629  frege93  39787  or2expropbilem1  42783
  Copyright terms: Public domain W3C validator