Metamath Proof Explorer


Theorem ichcircshi

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
Hypotheses ichcircshi.1 ( 𝑥 = 𝑧 → ( 𝜑𝜓 ) )
ichcircshi.2 ( 𝑦 = 𝑥 → ( 𝜓𝜒 ) )
ichcircshi.3 ( 𝑧 = 𝑦 → ( 𝜒𝜑 ) )
Assertion ichcircshi [ 𝑥𝑦 ] 𝜑

Proof

Step Hyp Ref Expression
1 ichcircshi.1 ( 𝑥 = 𝑧 → ( 𝜑𝜓 ) )
2 ichcircshi.2 ( 𝑦 = 𝑥 → ( 𝜓𝜒 ) )
3 ichcircshi.3 ( 𝑧 = 𝑦 → ( 𝜒𝜑 ) )
4 3 bicomd ( 𝑧 = 𝑦 → ( 𝜑𝜒 ) )
5 4 equcoms ( 𝑦 = 𝑧 → ( 𝜑𝜒 ) )
6 5 sbievw ( [ 𝑧 / 𝑦 ] 𝜑𝜒 )
7 6 2sbbii ( [ 𝑥 / 𝑧 ] [ 𝑦 / 𝑥 ] [ 𝑧 / 𝑦 ] 𝜑 ↔ [ 𝑥 / 𝑧 ] [ 𝑦 / 𝑥 ] 𝜒 )
8 2 bicomd ( 𝑦 = 𝑥 → ( 𝜒𝜓 ) )
9 8 equcoms ( 𝑥 = 𝑦 → ( 𝜒𝜓 ) )
10 9 sbievw ( [ 𝑦 / 𝑥 ] 𝜒𝜓 )
11 10 sbbii ( [ 𝑥 / 𝑧 ] [ 𝑦 / 𝑥 ] 𝜒 ↔ [ 𝑥 / 𝑧 ] 𝜓 )
12 1 bicomd ( 𝑥 = 𝑧 → ( 𝜓𝜑 ) )
13 12 equcoms ( 𝑧 = 𝑥 → ( 𝜓𝜑 ) )
14 13 sbievw ( [ 𝑥 / 𝑧 ] 𝜓𝜑 )
15 7 11 14 3bitri ( [ 𝑥 / 𝑧 ] [ 𝑦 / 𝑥 ] [ 𝑧 / 𝑦 ] 𝜑𝜑 )
16 15 gen2 𝑥𝑦 ( [ 𝑥 / 𝑧 ] [ 𝑦 / 𝑥 ] [ 𝑧 / 𝑦 ] 𝜑𝜑 )
17 df-ich ( [ 𝑥𝑦 ] 𝜑 ↔ ∀ 𝑥𝑦 ( [ 𝑥 / 𝑧 ] [ 𝑦 / 𝑥 ] [ 𝑧 / 𝑦 ] 𝜑𝜑 ) )
18 16 17 mpbir [ 𝑥𝑦 ] 𝜑