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Mirrors > Home > ILE Home > Th. List > sb9 | GIF version |
Description: Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 23-Mar-2018.) |
Ref | Expression |
---|---|
sb9 | ⊢ (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb9v 1854 | . . 3 ⊢ (∀𝑦[𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ ∀𝑤[𝑤 / 𝑦][𝑤 / 𝑥]𝜑) | |
2 | sbcom 1849 | . . . 4 ⊢ ([𝑤 / 𝑦][𝑤 / 𝑥]𝜑 ↔ [𝑤 / 𝑥][𝑤 / 𝑦]𝜑) | |
3 | 2 | albii 1359 | . . 3 ⊢ (∀𝑤[𝑤 / 𝑦][𝑤 / 𝑥]𝜑 ↔ ∀𝑤[𝑤 / 𝑥][𝑤 / 𝑦]𝜑) |
4 | sb9v 1854 | . . 3 ⊢ (∀𝑤[𝑤 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∀𝑥[𝑥 / 𝑤][𝑤 / 𝑦]𝜑) | |
5 | 1, 3, 4 | 3bitri 195 | . 2 ⊢ (∀𝑦[𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ ∀𝑥[𝑥 / 𝑤][𝑤 / 𝑦]𝜑) |
6 | ax-17 1419 | . . . 4 ⊢ (𝜑 → ∀𝑤𝜑) | |
7 | 6 | sbco2h 1838 | . . 3 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
8 | 7 | albii 1359 | . 2 ⊢ (∀𝑦[𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑) |
9 | 6 | sbco2h 1838 | . . 3 ⊢ ([𝑥 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑥 / 𝑦]𝜑) |
10 | 9 | albii 1359 | . 2 ⊢ (∀𝑥[𝑥 / 𝑤][𝑤 / 𝑦]𝜑 ↔ ∀𝑥[𝑥 / 𝑦]𝜑) |
11 | 5, 8, 10 | 3bitr3ri 200 | 1 ⊢ (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 ∀wal 1241 [wsb 1645 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 |
This theorem is referenced by: sb9i 1856 |
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