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Theorem sbco 1842
 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 1669 . . 3 [𝑦 / 𝑥]𝑦 = 𝑥
2 sbequ12 1654 . . . . 5 (𝑦 = 𝑥 → (𝜑 ↔ [𝑥 / 𝑦]𝜑))
32bicomd 129 . . . 4 (𝑦 = 𝑥 → ([𝑥 / 𝑦]𝜑𝜑))
43sbimi 1647 . . 3 ([𝑦 / 𝑥]𝑦 = 𝑥 → [𝑦 / 𝑥]([𝑥 / 𝑦]𝜑𝜑))
51, 4ax-mp 7 . 2 [𝑦 / 𝑥]([𝑥 / 𝑦]𝜑𝜑)
6 sbbi 1833 . 2 ([𝑦 / 𝑥]([𝑥 / 𝑦]𝜑𝜑) ↔ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑))
75, 6mpbi 133 1 ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
 Colors of variables: wff set class Syntax hints:   ↔ wb 98  [wsb 1645 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646 This theorem is referenced by:  sbco3v  1843
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