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Mirrors > Home > MPE Home > Th. List > Mathboxes > ichcom | Structured version Visualization version GIF version |
Description: The interchangeability of setvar variables is commutative. (Contributed by AV, 20-Aug-2023.) |
Ref | Expression |
---|---|
ichcom | ⊢ ([𝑥⇄𝑦]𝜓 ↔ [𝑦⇄𝑥]𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alcom 2163 | . . 3 ⊢ (∀𝑏∀𝑎([𝑏 / 𝑥][𝑎 / 𝑦]𝜓 ↔ [𝑎 / 𝑥][𝑏 / 𝑦]𝜓) ↔ ∀𝑎∀𝑏([𝑏 / 𝑥][𝑎 / 𝑦]𝜓 ↔ [𝑎 / 𝑥][𝑏 / 𝑦]𝜓)) | |
2 | sbcom2 2168 | . . . . 5 ⊢ ([𝑏 / 𝑥][𝑎 / 𝑦]𝜓 ↔ [𝑎 / 𝑦][𝑏 / 𝑥]𝜓) | |
3 | sbcom2 2168 | . . . . 5 ⊢ ([𝑎 / 𝑥][𝑏 / 𝑦]𝜓 ↔ [𝑏 / 𝑦][𝑎 / 𝑥]𝜓) | |
4 | 2, 3 | bibi12i 342 | . . . 4 ⊢ (([𝑏 / 𝑥][𝑎 / 𝑦]𝜓 ↔ [𝑎 / 𝑥][𝑏 / 𝑦]𝜓) ↔ ([𝑎 / 𝑦][𝑏 / 𝑥]𝜓 ↔ [𝑏 / 𝑦][𝑎 / 𝑥]𝜓)) |
5 | 4 | 2albii 1821 | . . 3 ⊢ (∀𝑎∀𝑏([𝑏 / 𝑥][𝑎 / 𝑦]𝜓 ↔ [𝑎 / 𝑥][𝑏 / 𝑦]𝜓) ↔ ∀𝑎∀𝑏([𝑎 / 𝑦][𝑏 / 𝑥]𝜓 ↔ [𝑏 / 𝑦][𝑎 / 𝑥]𝜓)) |
6 | 1, 5 | bitri 277 | . 2 ⊢ (∀𝑏∀𝑎([𝑏 / 𝑥][𝑎 / 𝑦]𝜓 ↔ [𝑎 / 𝑥][𝑏 / 𝑦]𝜓) ↔ ∀𝑎∀𝑏([𝑎 / 𝑦][𝑏 / 𝑥]𝜓 ↔ [𝑏 / 𝑦][𝑎 / 𝑥]𝜓)) |
7 | dfich2 43633 | . 2 ⊢ ([𝑥⇄𝑦]𝜓 ↔ ∀𝑏∀𝑎([𝑏 / 𝑥][𝑎 / 𝑦]𝜓 ↔ [𝑎 / 𝑥][𝑏 / 𝑦]𝜓)) | |
8 | dfich2 43633 | . 2 ⊢ ([𝑦⇄𝑥]𝜓 ↔ ∀𝑎∀𝑏([𝑎 / 𝑦][𝑏 / 𝑥]𝜓 ↔ [𝑏 / 𝑦][𝑎 / 𝑥]𝜓)) | |
9 | 6, 7, 8 | 3bitr4i 305 | 1 ⊢ ([𝑥⇄𝑦]𝜓 ↔ [𝑦⇄𝑥]𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∀wal 1535 [wsb 2069 [wich 43625 |
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-10 2145 ax-11 2161 ax-12 2177 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-ich 43626 |
This theorem is referenced by: ichnfb 43645 ich2exprop 43653 |
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