![]() |
Mathbox for BJ |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-19.12 | Structured version Visualization version GIF version |
Description: See 19.12 2335. Could be labeled "exalimalex" for "'there exists for all' implies 'for all there exists'". This proof is from excom 2166 and modal (B) on top of modalK logic. (Contributed by BJ, 12-Aug-2023.) The proof should not rely on df-nf 1786 or df-bj-nnf 34171, directly or indirectly. (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-19.12 | ⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-modalbe 34135 | . 2 ⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑦∃𝑥∀𝑦𝜑) | |
2 | excom 2166 | . . 3 ⊢ (∃𝑦∃𝑥∀𝑦𝜑 ↔ ∃𝑥∃𝑦∀𝑦𝜑) | |
3 | axc7e 2326 | . . . 4 ⊢ (∃𝑦∀𝑦𝜑 → 𝜑) | |
4 | 3 | eximi 1836 | . . 3 ⊢ (∃𝑥∃𝑦∀𝑦𝜑 → ∃𝑥𝜑) |
5 | 2, 4 | sylbi 220 | . 2 ⊢ (∃𝑦∃𝑥∀𝑦𝜑 → ∃𝑥𝜑) |
6 | 1, 5 | sylg 1824 | 1 ⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1536 ∃wex 1781 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2142 ax-11 2158 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-ex 1782 |
This theorem is referenced by: bj-nnflemae 34208 bj-nnflemea 34209 |
Copyright terms: Public domain | W3C validator |