![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 2nexaln | Structured version Visualization version GIF version |
Description: Theorem *11.25 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.) |
Ref | Expression |
---|---|
2nexaln | ⊢ (¬ ∃𝑥∃𝑦𝜑 ↔ ∀𝑥∀𝑦 ¬ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2exnaln 1830 | . . 3 ⊢ (∃𝑥∃𝑦𝜑 ↔ ¬ ∀𝑥∀𝑦 ¬ 𝜑) | |
2 | 1 | bicomi 227 | . 2 ⊢ (¬ ∀𝑥∀𝑦 ¬ 𝜑 ↔ ∃𝑥∃𝑦𝜑) |
3 | 2 | con1bii 360 | 1 ⊢ (¬ ∃𝑥∃𝑦𝜑 ↔ ∀𝑥∀𝑦 ¬ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 ∀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 |
This theorem depends on definitions: df-bi 210 df-ex 1782 |
This theorem is referenced by: cbvex2 2423 2mo 2710 bj-alcomexcom 34127 pm11.63 41099 fun2dmnopgexmpl 43840 spr0nelg 43993 |
Copyright terms: Public domain | W3C validator |