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Theorem sbco3 1947
Description: A composition law for substitution. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 22-Mar-2018.)
Assertion
Ref Expression
sbco3 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑)

Proof of Theorem sbco3
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 sbco3xzyz 1946 . . 3 ([𝑤 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑤 / 𝑥][𝑥 / 𝑦]𝜑)
21sbbii 1738 . 2 ([𝑧 / 𝑤][𝑤 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑤][𝑤 / 𝑥][𝑥 / 𝑦]𝜑)
3 ax-17 1506 . . 3 ([𝑦 / 𝑥]𝜑 → ∀𝑤[𝑦 / 𝑥]𝜑)
43sbco2h 1937 . 2 ([𝑧 / 𝑤][𝑤 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑)
5 ax-17 1506 . . 3 ([𝑥 / 𝑦]𝜑 → ∀𝑤[𝑥 / 𝑦]𝜑)
65sbco2h 1937 . 2 ([𝑧 / 𝑤][𝑤 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑)
72, 4, 63bitr3i 209 1 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑)
Colors of variables: wff set class
Syntax hints:  wb 104  [wsb 1735
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736
This theorem is referenced by:  sbcom  1948
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