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Theorem sbco 1858
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 1685 . . 3 [𝑦 / 𝑥]𝑦 = 𝑥
2 sbequ12 1670 . . . . 5 (𝑦 = 𝑥 → (𝜑 ↔ [𝑥 / 𝑦]𝜑))
32bicomd 133 . . . 4 (𝑦 = 𝑥 → ([𝑥 / 𝑦]𝜑𝜑))
43sbimi 1663 . . 3 ([𝑦 / 𝑥]𝑦 = 𝑥 → [𝑦 / 𝑥]([𝑥 / 𝑦]𝜑𝜑))
51, 4ax-mp 7 . 2 [𝑦 / 𝑥]([𝑥 / 𝑦]𝜑𝜑)
6 sbbi 1849 . 2 ([𝑦 / 𝑥]([𝑥 / 𝑦]𝜑𝜑) ↔ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑))
75, 6mpbi 137 1 ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
Colors of variables: wff set class
Syntax hints:  wb 102  [wsb 1661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662
This theorem is referenced by:  sbco3v  1859
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