Theorem ichcircshi 44794

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 → (𝜑 ↔ 𝜓))
ichcircshi.2	⊢ (𝑦 = 𝑥 → (𝜓 ↔ 𝜒))
ichcircshi.3	⊢ (𝑧 = 𝑦 → (𝜒 ↔ 𝜑))

Assertion

Ref	Expression
ichcircshi	⊢ [𝑥⇄𝑦]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

Step	Hyp	Ref	Expression
1		ichcircshi.3	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝜒 ↔ 𝜑))
2	1	bicomd 222	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜒))
3	2	equcoms 2024	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜒))
4	3	sbievw 2097	. . . . 5 ⊢ ([𝑧 / 𝑦]𝜑 ↔ 𝜒)
5	4	2sbbii 2081	. . . 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 2024	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜒 ↔ 𝜓))
9	8	sbievw 2097	. . . . 5 ⊢ ([𝑦 / 𝑥]𝜒 ↔ 𝜓)
10	9	sbbii 2080	. . . 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 2024	. . . . 5 ⊢ (𝑧 = 𝑥 → (𝜓 ↔ 𝜑))
14	13	sbievw 2097	. . . 4 ⊢ ([𝑥 / 𝑧]𝜓 ↔ 𝜑)
15	5, 10, 14	3bitri 296	. . 3 ⊢ ([𝑥 / 𝑧][𝑦 / 𝑥][𝑧 / 𝑦]𝜑 ↔ 𝜑)
16	15	gen2 1800	. 2 ⊢ ∀𝑥∀𝑦([𝑥 / 𝑧][𝑦 / 𝑥][𝑧 / 𝑦]𝜑 ↔ 𝜑)
17		df-ich 44786	. 2 ⊢ ([𝑥⇄𝑦]𝜑 ↔ ∀𝑥∀𝑦([𝑥 / 𝑧][𝑦 / 𝑥][𝑧 / 𝑦]𝜑 ↔ 𝜑))
18	16, 17	mpbir 230	1 ⊢ [𝑥⇄𝑦]𝜑

Syntax hints:   ↔ wb 205  ∀wal 1537  [wsb 2068  [wich 44785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-sb 2069  df-ich 44786

This theorem is referenced by:  ichexmpl1  44809  ichexmpl2  44810