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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > df-ich | Structured version Visualization version GIF version |
Description: Define the property of a wff 𝜑 that the setvar variables 𝑥 and 𝑦 are interchangeable. For an alternate definition using implicit substitution and a temporary setvar variable see ichcircshi 46667. Another, equivalent definition using two temporary setvar variables is provided in dfich2 46671. (Contributed by AV, 29-Jul-2023.) |
Ref | Expression |
---|---|
df-ich | ⊢ ([𝑥⇄𝑦]𝜑 ↔ ∀𝑥∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wph | . . 3 wff 𝜑 | |
2 | vx | . . 3 setvar 𝑥 | |
3 | vy | . . 3 setvar 𝑦 | |
4 | 1, 2, 3 | wich 46658 | . 2 wff [𝑥⇄𝑦]𝜑 |
5 | va | . . . . . . . 8 setvar 𝑎 | |
6 | 1, 3, 5 | wsb 2059 | . . . . . . 7 wff [𝑎 / 𝑦]𝜑 |
7 | 6, 2, 3 | wsb 2059 | . . . . . 6 wff [𝑦 / 𝑥][𝑎 / 𝑦]𝜑 |
8 | 7, 5, 2 | wsb 2059 | . . . . 5 wff [𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 |
9 | 8, 1 | wb 205 | . . . 4 wff ([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑) |
10 | 9, 3 | wal 1531 | . . 3 wff ∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑) |
11 | 10, 2 | wal 1531 | . 2 wff ∀𝑥∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑) |
12 | 4, 11 | wb 205 | 1 wff ([𝑥⇄𝑦]𝜑 ↔ ∀𝑥∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑)) |
Colors of variables: wff setvar class |
This definition is referenced by: nfich1 46660 nfich2 46661 ichv 46662 ichf 46663 ichid 46664 icht 46665 ichbidv 46666 ichcircshi 46667 ichan 46668 ichn 46669 dfich2 46671 icheq 46675 ichal 46679 ich2exprop 46684 |
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