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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-alcomexcom | Structured version Visualization version GIF version |
Description: Commutation of two existential quantifiers on a formula is equivalent to commutation of two universal quantifiers over the same variables on the negation of that formula. Can be placed in the ax-4 1812 section, soon after 2nexaln 1833, and used to prove excom 2163. (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 | con34b 316 | . 2 ⊢ ((∃𝑦∃𝑥𝜑 → ∃𝑥∃𝑦𝜑) ↔ (¬ ∃𝑥∃𝑦𝜑 → ¬ ∃𝑦∃𝑥𝜑)) | |
2 | 2nexaln 1833 | . . 3 ⊢ (¬ ∃𝑥∃𝑦𝜑 ↔ ∀𝑥∀𝑦 ¬ 𝜑) | |
3 | 2nexaln 1833 | . . 3 ⊢ (¬ ∃𝑦∃𝑥𝜑 ↔ ∀𝑦∀𝑥 ¬ 𝜑) | |
4 | 2, 3 | imbi12i 351 | . 2 ⊢ ((¬ ∃𝑥∃𝑦𝜑 → ¬ ∃𝑦∃𝑥𝜑) ↔ (∀𝑥∀𝑦 ¬ 𝜑 → ∀𝑦∀𝑥 ¬ 𝜑)) |
5 | 1, 4 | bitr2i 276 | 1 ⊢ ((∀𝑥∀𝑦 ¬ 𝜑 → ∀𝑦∀𝑥 ¬ 𝜑) ↔ (∃𝑦∃𝑥𝜑 → ∃𝑥∃𝑦𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∀wal 1540 ∃wex 1782 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 |
This theorem depends on definitions: df-bi 206 df-ex 1783 |
This theorem is referenced by: (None) |
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