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Theorem ichcircshi 43910
Description: The setvar variables are interchangeable if they can be circularily shifted using a third setvar variable, using implicit substitution. (Contributed by AV, 29-Jul-2023.)
Hypotheses
Ref Expression
ichcircshi.1 (𝑥 = 𝑧 → (𝜑𝜓))
ichcircshi.2 (𝑦 = 𝑥 → (𝜓𝜒))
ichcircshi.3 (𝑧 = 𝑦 → (𝜒𝜑))
Assertion
Ref Expression
ichcircshi [𝑥𝑦]𝜑
Distinct variable groups:   𝜑,𝑧   𝜓,𝑥   𝜒,𝑦   𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦,𝑧)   𝜒(𝑥,𝑧)

Proof of Theorem ichcircshi
StepHypRef Expression
1 ichcircshi.3 . . . . . . . 8 (𝑧 = 𝑦 → (𝜒𝜑))
21bicomd 226 . . . . . . 7 (𝑧 = 𝑦 → (𝜑𝜒))
32equcoms 2027 . . . . . 6 (𝑦 = 𝑧 → (𝜑𝜒))
43sbievw 2103 . . . . 5 ([𝑧 / 𝑦]𝜑𝜒)
542sbbii 2082 . . . 4 ([𝑥 / 𝑧][𝑦 / 𝑥][𝑧 / 𝑦]𝜑 ↔ [𝑥 / 𝑧][𝑦 / 𝑥]𝜒)
6 ichcircshi.2 . . . . . . . 8 (𝑦 = 𝑥 → (𝜓𝜒))
76bicomd 226 . . . . . . 7 (𝑦 = 𝑥 → (𝜒𝜓))
87equcoms 2027 . . . . . 6 (𝑥 = 𝑦 → (𝜒𝜓))
98sbievw 2103 . . . . 5 ([𝑦 / 𝑥]𝜒𝜓)
109sbbii 2081 . . . 4 ([𝑥 / 𝑧][𝑦 / 𝑥]𝜒 ↔ [𝑥 / 𝑧]𝜓)
11 ichcircshi.1 . . . . . . 7 (𝑥 = 𝑧 → (𝜑𝜓))
1211bicomd 226 . . . . . 6 (𝑥 = 𝑧 → (𝜓𝜑))
1312equcoms 2027 . . . . 5 (𝑧 = 𝑥 → (𝜓𝜑))
1413sbievw 2103 . . . 4 ([𝑥 / 𝑧]𝜓𝜑)
155, 10, 143bitri 300 . . 3 ([𝑥 / 𝑧][𝑦 / 𝑥][𝑧 / 𝑦]𝜑𝜑)
1615gen2 1798 . 2 𝑥𝑦([𝑥 / 𝑧][𝑦 / 𝑥][𝑧 / 𝑦]𝜑𝜑)
17 df-ich 43902 . 2 ([𝑥𝑦]𝜑 ↔ ∀𝑥𝑦([𝑥 / 𝑧][𝑦 / 𝑥][𝑧 / 𝑦]𝜑𝜑))
1816, 17mpbir 234 1 [𝑥𝑦]𝜑
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1536  [wsb 2069  [wich 43901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-ich 43902
This theorem is referenced by:  ichexmpl1  43925  ichexmpl2  43926
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