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Theorem sbco 1919
Description: A composition law for substitution. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
sbco ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Proof of Theorem sbco
StepHypRef Expression
1 equsb2 1744 . . 3 [𝑦 / 𝑥]𝑦 = 𝑥
2 sbequ12 1729 . . . . 5 (𝑦 = 𝑥 → (𝜑 ↔ [𝑥 / 𝑦]𝜑))
32bicomd 140 . . . 4 (𝑦 = 𝑥 → ([𝑥 / 𝑦]𝜑𝜑))
43sbimi 1722 . . 3 ([𝑦 / 𝑥]𝑦 = 𝑥 → [𝑦 / 𝑥]([𝑥 / 𝑦]𝜑𝜑))
51, 4ax-mp 5 . 2 [𝑦 / 𝑥]([𝑥 / 𝑦]𝜑𝜑)
6 sbbi 1910 . 2 ([𝑦 / 𝑥]([𝑥 / 𝑦]𝜑𝜑) ↔ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑))
75, 6mpbi 144 1 ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
Colors of variables: wff set class
Syntax hints:  wb 104  [wsb 1720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721
This theorem is referenced by:  sbco3v  1920
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