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Theorem sbco 2542
Description: A composition law for substitution. See sbcov 2248 for a version with a disjoint variable condition requiring fewer axioms. (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 2541 . 2 ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑦]𝜑)
2 sbid 2247 . . 3 ([𝑦 / 𝑦]𝜑𝜑)
32sbbii 2072 . 2 ([𝑦 / 𝑥][𝑦 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
41, 3bitri 276 1 ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 207  [wsb 2060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-10 2136  ax-12 2167  ax-13 2381
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-ex 1772  df-nf 1776  df-sb 2061
This theorem is referenced by:  sbid2  2543  sbco3  2548
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