Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ichid Structured version   Visualization version   GIF version

Theorem ichid 45733
Description: A setvar variable is always interchangeable with itself. (Contributed by AV, 29-Jul-2023.)
Assertion
Ref Expression
ichid [𝑥𝑥]𝜑

Proof of Theorem ichid
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 sbid 2248 . . . . 5 ([𝑥 / 𝑥][𝑎 / 𝑥]𝜑 ↔ [𝑎 / 𝑥]𝜑)
21sbbii 2080 . . . 4 ([𝑥 / 𝑎][𝑥 / 𝑥][𝑎 / 𝑥]𝜑 ↔ [𝑥 / 𝑎][𝑎 / 𝑥]𝜑)
3 sbid2vw 2251 . . . 4 ([𝑥 / 𝑎][𝑎 / 𝑥]𝜑𝜑)
42, 3bitri 275 . . 3 ([𝑥 / 𝑎][𝑥 / 𝑥][𝑎 / 𝑥]𝜑𝜑)
54gen2 1799 . 2 𝑥𝑥([𝑥 / 𝑎][𝑥 / 𝑥][𝑎 / 𝑥]𝜑𝜑)
6 df-ich 45728 . 2 ([𝑥𝑥]𝜑 ↔ ∀𝑥𝑥([𝑥 / 𝑎][𝑥 / 𝑥][𝑎 / 𝑥]𝜑𝜑))
75, 6mpbir 230 1 [𝑥𝑥]𝜑
Colors of variables: wff setvar class
Syntax hints:  wb 205  wal 1540  [wsb 2068  [wich 45727
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-12 2172
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-sb 2069  df-ich 45728
This theorem is referenced by:  icheqid  45743
  Copyright terms: Public domain W3C validator