Metamath Proof Explorer


Definition df-ich

Description: Define the property of a wff ph that the setvar variables x and y are interchangeable. For an alternate definition using implicit substitution and a temporary setvar variable see ichcircshi . Another, equivalent definition using two temporary setvar variables is provided in dfich2 . (Contributed by AV, 29-Jul-2023)

Ref Expression
Assertion df-ich ( [ 𝑥𝑦 ] 𝜑 ↔ ∀ 𝑥𝑦 ( [ 𝑥 / 𝑎 ] [ 𝑦 / 𝑥 ] [ 𝑎 / 𝑦 ] 𝜑𝜑 ) )

Detailed syntax breakdown

Step Hyp Ref Expression
0 vx 𝑥
1 vy 𝑦
2 wph 𝜑
3 2 0 1 wich [ 𝑥𝑦 ] 𝜑
4 va 𝑎
5 2 1 4 wsb [ 𝑎 / 𝑦 ] 𝜑
6 5 0 1 wsb [ 𝑦 / 𝑥 ] [ 𝑎 / 𝑦 ] 𝜑
7 6 4 0 wsb [ 𝑥 / 𝑎 ] [ 𝑦 / 𝑥 ] [ 𝑎 / 𝑦 ] 𝜑
8 7 2 wb ( [ 𝑥 / 𝑎 ] [ 𝑦 / 𝑥 ] [ 𝑎 / 𝑦 ] 𝜑𝜑 )
9 8 1 wal 𝑦 ( [ 𝑥 / 𝑎 ] [ 𝑦 / 𝑥 ] [ 𝑎 / 𝑦 ] 𝜑𝜑 )
10 9 0 wal 𝑥𝑦 ( [ 𝑥 / 𝑎 ] [ 𝑦 / 𝑥 ] [ 𝑎 / 𝑦 ] 𝜑𝜑 )
11 3 10 wb ( [ 𝑥𝑦 ] 𝜑 ↔ ∀ 𝑥𝑦 ( [ 𝑥 / 𝑎 ] [ 𝑦 / 𝑥 ] [ 𝑎 / 𝑦 ] 𝜑𝜑 ) )