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Mirrors > Home > MPE Home > Th. List > Mathboxes > icht | Structured version Visualization version GIF version |
Description: A theorem is interchangeable. (Contributed by SN, 4-May-2024.) |
Ref | Expression |
---|---|
icht.1 | ⊢ 𝜑 |
Ref | Expression |
---|---|
icht | ⊢ [𝑥⇄𝑦]𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | icht.1 | . . . . . . 7 ⊢ 𝜑 | |
2 | 1 | sbt 2072 | . . . . . 6 ⊢ [𝑎 / 𝑦]𝜑 |
3 | 2 | sbt 2072 | . . . . 5 ⊢ [𝑦 / 𝑥][𝑎 / 𝑦]𝜑 |
4 | 3 | sbt 2072 | . . . 4 ⊢ [𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 |
5 | 4, 1 | 2th 267 | . . 3 ⊢ ([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑) |
6 | 5 | gen2 1799 | . 2 ⊢ ∀𝑥∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑) |
7 | df-ich 44316 | . 2 ⊢ ([𝑥⇄𝑦]𝜑 ↔ ∀𝑥∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑)) | |
8 | 6, 7 | mpbir 234 | 1 ⊢ [𝑥⇄𝑦]𝜑 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∀wal 1537 [wsb 2070 [wich 44315 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 |
This theorem depends on definitions: df-bi 210 df-sb 2071 df-ich 44316 |
This theorem is referenced by: (None) |
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