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Theorem sbco 2400
 Description: A composition law for substitution. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 21-Sep-2018.)
Assertion
Ref Expression
sbco ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Proof of Theorem sbco
StepHypRef Expression
1 sbcom3 2399 . 2 ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑦]𝜑)
2 sbid 2100 . . 3 ([𝑦 / 𝑦]𝜑𝜑)
32sbbii 1874 . 2 ([𝑦 / 𝑥][𝑦 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
41, 3bitri 263 1 ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195  [wsb 1867 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868 This theorem is referenced by:  sbid2  2401  sbco3  2405  sb6a  2436
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