Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 1833 | . . 3 ⊢ (∃𝑦 ¬ 𝜑 ↔ ¬ ∀𝑦𝜑) | |
2 | 1 | albii 1826 | . 2 ⊢ (∀𝑥∃𝑦 ¬ 𝜑 ↔ ∀𝑥 ¬ ∀𝑦𝜑) |
3 | alnex 1788 | . 2 ⊢ (∀𝑥 ¬ ∀𝑦𝜑 ↔ ¬ ∃𝑥∀𝑦𝜑) | |
4 | 2, 3 | bitri 278 | 1 ⊢ (∀𝑥∃𝑦 ¬ 𝜑 ↔ ¬ ∃𝑥∀𝑦𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 ∀wal 1540 ∃wex 1786 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 |
This theorem depends on definitions: df-bi 210 df-ex 1787 |
This theorem is referenced by: 2exnexn 1852 nalset 5178 kmlem2 9644 |
Copyright terms: Public domain | W3C validator |