Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbid2 Structured version   Visualization version   GIF version

Theorem sbid2 2412
 Description: An identity law for substitution. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)
Hypothesis
Ref Expression
sbid2.1 𝑥𝜑
Assertion
Ref Expression
sbid2 ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑𝜑)

Proof of Theorem sbid2
StepHypRef Expression
1 sbco 2411 . 2 ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
2 sbid2.1 . . 3 𝑥𝜑
32sbf 2379 . 2 ([𝑦 / 𝑥]𝜑𝜑)
41, 3bitri 264 1 ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑𝜑)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196  Ⅎwnf 1705  [wsb 1877 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-10 2016  ax-12 2044  ax-13 2245 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1702  df-nf 1707  df-sb 1878 This theorem is referenced by:  sbtrt  2419  sbid2v  2456
 Copyright terms: Public domain W3C validator