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Theorem sbcov 2255
 Description: A composition law for substitution. Version of sbco 2526 with a disjoint variable condition using fewer axioms. (Contributed by NM, 14-May-1993.) (Revised by Gino Giotto, 7-Aug-2023.)
Assertion
Ref Expression
sbcov ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sbcov
StepHypRef Expression
1 sbcom3vv 2103 . 2 ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑦]𝜑)
2 sbid 2254 . . 3 ([𝑦 / 𝑦]𝜑𝜑)
32sbbii 2081 . 2 ([𝑦 / 𝑥][𝑦 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
41, 3bitri 278 1 ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
 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 2175 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070 This theorem is referenced by:  sb6a  2256  sbbibOLD  2285
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