Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ichid | Structured version Visualization version GIF version |
Description: A setvar variable is always interchangeable with itself. (Contributed by AV, 29-Jul-2023.) |
Ref | Expression |
---|---|
ichid | ⊢ [𝑥⇄𝑥]𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbid 2257 | . . . . 5 ⊢ ([𝑥 / 𝑥][𝑎 / 𝑥]𝜑 ↔ [𝑎 / 𝑥]𝜑) | |
2 | 1 | sbbii 2081 | . . . 4 ⊢ ([𝑥 / 𝑎][𝑥 / 𝑥][𝑎 / 𝑥]𝜑 ↔ [𝑥 / 𝑎][𝑎 / 𝑥]𝜑) |
3 | sbid2vw 2260 | . . . 4 ⊢ ([𝑥 / 𝑎][𝑎 / 𝑥]𝜑 ↔ 𝜑) | |
4 | 2, 3 | bitri 277 | . . 3 ⊢ ([𝑥 / 𝑎][𝑥 / 𝑥][𝑎 / 𝑥]𝜑 ↔ 𝜑) |
5 | 4 | gen2 1797 | . 2 ⊢ ∀𝑥∀𝑥([𝑥 / 𝑎][𝑥 / 𝑥][𝑎 / 𝑥]𝜑 ↔ 𝜑) |
6 | df-ich 43655 | . 2 ⊢ ([𝑥⇄𝑥]𝜑 ↔ ∀𝑥∀𝑥([𝑥 / 𝑎][𝑥 / 𝑥][𝑎 / 𝑥]𝜑 ↔ 𝜑)) | |
7 | 5, 6 | mpbir 233 | 1 ⊢ [𝑥⇄𝑥]𝜑 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∀wal 1535 [wsb 2069 [wich 43654 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-12 2177 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-sb 2070 df-ich 43655 |
This theorem is referenced by: icheqid 43668 |
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