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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-alcomexcom | Structured version Visualization version GIF version |
Description: Commutation of universal quantifiers implies commutation of existential quantifiers. Can be placed in the ax-4 1853 section, soon after 2nexaln 1873, and used to prove excom 2155. (Contributed by BJ, 29-Nov-2020.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-alcomexcom | ⊢ ((∀𝑥∀𝑦 ¬ 𝜑 → ∀𝑦∀𝑥 ¬ 𝜑) → (∃𝑦∃𝑥𝜑 → ∃𝑥∃𝑦𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2nexaln 1873 | . . 3 ⊢ (¬ ∃𝑥∃𝑦𝜑 ↔ ∀𝑥∀𝑦 ¬ 𝜑) | |
2 | 2nexaln 1873 | . . 3 ⊢ (¬ ∃𝑦∃𝑥𝜑 ↔ ∀𝑦∀𝑥 ¬ 𝜑) | |
3 | 1, 2 | imbi12i 342 | . 2 ⊢ ((¬ ∃𝑥∃𝑦𝜑 → ¬ ∃𝑦∃𝑥𝜑) ↔ (∀𝑥∀𝑦 ¬ 𝜑 → ∀𝑦∀𝑥 ¬ 𝜑)) |
4 | con4 113 | . 2 ⊢ ((¬ ∃𝑥∃𝑦𝜑 → ¬ ∃𝑦∃𝑥𝜑) → (∃𝑦∃𝑥𝜑 → ∃𝑥∃𝑦𝜑)) | |
5 | 3, 4 | sylbir 227 | 1 ⊢ ((∀𝑥∀𝑦 ¬ 𝜑 → ∀𝑦∀𝑥 ¬ 𝜑) → (∃𝑦∃𝑥𝜑 → ∃𝑥∃𝑦𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1599 ∃wex 1823 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 |
This theorem depends on definitions: df-bi 199 df-ex 1824 |
This theorem is referenced by: (None) |
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