Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ichcircshi | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
ichcircshi.1 | ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓)) |
ichcircshi.2 | ⊢ (𝑦 = 𝑥 → (𝜓 ↔ 𝜒)) |
ichcircshi.3 | ⊢ (𝑧 = 𝑦 → (𝜒 ↔ 𝜑)) |
Ref | Expression |
---|---|
ichcircshi | ⊢ [𝑥⇄𝑦]𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ichcircshi.3 | . . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝜒 ↔ 𝜑)) | |
2 | 1 | bicomd 222 | . . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜒)) |
3 | 2 | equcoms 2023 | . . . . . 6 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜒)) |
4 | 3 | sbievw 2095 | . . . . 5 ⊢ ([𝑧 / 𝑦]𝜑 ↔ 𝜒) |
5 | 4 | 2sbbii 2080 | . . . 4 ⊢ ([𝑥 / 𝑧][𝑦 / 𝑥][𝑧 / 𝑦]𝜑 ↔ [𝑥 / 𝑧][𝑦 / 𝑥]𝜒) |
6 | ichcircshi.2 | . . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝜓 ↔ 𝜒)) | |
7 | 6 | bicomd 222 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝜒 ↔ 𝜓)) |
8 | 7 | equcoms 2023 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜒 ↔ 𝜓)) |
9 | 8 | sbievw 2095 | . . . . 5 ⊢ ([𝑦 / 𝑥]𝜒 ↔ 𝜓) |
10 | 9 | sbbii 2079 | . . . 4 ⊢ ([𝑥 / 𝑧][𝑦 / 𝑥]𝜒 ↔ [𝑥 / 𝑧]𝜓) |
11 | ichcircshi.1 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓)) | |
12 | 11 | bicomd 222 | . . . . . 6 ⊢ (𝑥 = 𝑧 → (𝜓 ↔ 𝜑)) |
13 | 12 | equcoms 2023 | . . . . 5 ⊢ (𝑧 = 𝑥 → (𝜓 ↔ 𝜑)) |
14 | 13 | sbievw 2095 | . . . 4 ⊢ ([𝑥 / 𝑧]𝜓 ↔ 𝜑) |
15 | 5, 10, 14 | 3bitri 297 | . . 3 ⊢ ([𝑥 / 𝑧][𝑦 / 𝑥][𝑧 / 𝑦]𝜑 ↔ 𝜑) |
16 | 15 | gen2 1799 | . 2 ⊢ ∀𝑥∀𝑦([𝑥 / 𝑧][𝑦 / 𝑥][𝑧 / 𝑦]𝜑 ↔ 𝜑) |
17 | df-ich 44876 | . 2 ⊢ ([𝑥⇄𝑦]𝜑 ↔ ∀𝑥∀𝑦([𝑥 / 𝑧][𝑦 / 𝑥][𝑧 / 𝑦]𝜑 ↔ 𝜑)) | |
18 | 16, 17 | mpbir 230 | 1 ⊢ [𝑥⇄𝑦]𝜑 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∀wal 1537 [wsb 2067 [wich 44875 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1783 df-sb 2068 df-ich 44876 |
This theorem is referenced by: ichexmpl1 44899 ichexmpl2 44900 |
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