Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbid2vw Structured version   Visualization version   GIF version

Theorem sbid2vw 2261
 Description: Reverting substitution yields the original expression. Based on fewer axioms than sbid2v 2551, at the expense of an extra distinct variable condition. (Contributed by NM, 14-May-1993.) (Revised by Wolf Lammen, 5-Aug-2023.)
Assertion
Ref Expression
sbid2vw ([𝑡 / 𝑥][𝑥 / 𝑡]𝜑𝜑)
Distinct variable groups:   𝑥,𝑡   𝜑,𝑥
Allowed substitution hint:   𝜑(𝑡)

Proof of Theorem sbid2vw
StepHypRef Expression
1 sbequ12r 2255 . 2 (𝑥 = 𝑡 → ([𝑥 / 𝑡]𝜑𝜑))
21sbievw 2103 1 ([𝑡 / 𝑥][𝑥 / 𝑡]𝜑𝜑)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209  [wsb 2069 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2178 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070 This theorem is referenced by:  sbco4lem  2284  ichid  43907
 Copyright terms: Public domain W3C validator