| Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
| 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 2159 | . . 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 2173 | . . . . 5 ⊢ ([𝑏 / 𝑥][𝑎 / 𝑦]𝜓 ↔ [𝑎 / 𝑦][𝑏 / 𝑥]𝜓) | |
| 3 | sbcom2 2173 | . . . . 5 ⊢ ([𝑎 / 𝑥][𝑏 / 𝑦]𝜓 ↔ [𝑏 / 𝑦][𝑎 / 𝑥]𝜓) | |
| 4 | 2, 3 | bibi12i 339 | . . . 4 ⊢ (([𝑏 / 𝑥][𝑎 / 𝑦]𝜓 ↔ [𝑎 / 𝑥][𝑏 / 𝑦]𝜓) ↔ ([𝑎 / 𝑦][𝑏 / 𝑥]𝜓 ↔ [𝑏 / 𝑦][𝑎 / 𝑥]𝜓)) |
| 5 | 4 | 2albii 1820 | . . 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 275 | . 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 47445 | . 2 ⊢ ([𝑥⇄𝑦]𝜓 ↔ ∀𝑏∀𝑎([𝑏 / 𝑥][𝑎 / 𝑦]𝜓 ↔ [𝑎 / 𝑥][𝑏 / 𝑦]𝜓)) | |
| 8 | dfich2 47445 | . 2 ⊢ ([𝑦⇄𝑥]𝜓 ↔ ∀𝑎∀𝑏([𝑎 / 𝑦][𝑏 / 𝑥]𝜓 ↔ [𝑏 / 𝑦][𝑎 / 𝑥]𝜓)) | |
| 9 | 6, 7, 8 | 3bitr4i 303 | 1 ⊢ ([𝑥⇄𝑦]𝜓 ↔ [𝑦⇄𝑥]𝜓) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∀wal 1538 [wsb 2064 [wich 47432 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-10 2141 ax-11 2157 ax-12 2177 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-ich 47433 |
| This theorem is referenced by: ichnfb 47452 ich2exprop 47458 |
| Copyright terms: Public domain | W3C validator |