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Mirrors > Home > MPE Home > Th. List > alexn | Structured version Visualization version GIF version |
Description: A relationship between two quantifiers and negation. (Contributed by NM, 18-Aug-1993.) |
Ref | Expression |
---|---|
alexn | ⊢ (∀𝑥∃𝑦 ¬ 𝜑 ↔ ¬ ∃𝑥∀𝑦𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exnal 1825 | . . 3 ⊢ (∃𝑦 ¬ 𝜑 ↔ ¬ ∀𝑦𝜑) | |
2 | 1 | albii 1817 | . 2 ⊢ (∀𝑥∃𝑦 ¬ 𝜑 ↔ ∀𝑥 ¬ ∀𝑦𝜑) |
3 | alnex 1779 | . 2 ⊢ (∀𝑥 ¬ ∀𝑦𝜑 ↔ ¬ ∃𝑥∀𝑦𝜑) | |
4 | 2, 3 | bitri 274 | 1 ⊢ (∀𝑥∃𝑦 ¬ 𝜑 ↔ ¬ ∃𝑥∀𝑦𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∀wal 1535 ∃wex 1777 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 |
This theorem depends on definitions: df-bi 206 df-ex 1778 |
This theorem is referenced by: 2exnexn 1844 nalset 5240 kmlem2 9935 |
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