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| Mirrors > Home > MPE Home > Th. List > 2exnaln | Structured version Visualization version GIF version | ||
| Description: Theorem *11.22 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.) |
| Ref | Expression |
|---|---|
| 2exnaln | ⊢ (∃𝑥∃𝑦𝜑 ↔ ¬ ∀𝑥∀𝑦 ¬ 𝜑) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-ex 1780 | . 2 ⊢ (∃𝑥∃𝑦𝜑 ↔ ¬ ∀𝑥 ¬ ∃𝑦𝜑) | |
| 2 | alnex 1781 | . . 3 ⊢ (∀𝑦 ¬ 𝜑 ↔ ¬ ∃𝑦𝜑) | |
| 3 | 2 | albii 1819 | . 2 ⊢ (∀𝑥∀𝑦 ¬ 𝜑 ↔ ∀𝑥 ¬ ∃𝑦𝜑) |
| 4 | 1, 3 | xchbinxr 335 | 1 ⊢ (∃𝑥∃𝑦𝜑 ↔ ¬ ∀𝑥∀𝑦 ¬ 𝜑) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ↔ wb 206 ∀wal 1538 ∃wex 1779 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 |
| This theorem depends on definitions: df-bi 207 df-ex 1780 |
| This theorem is referenced by: 2nexaln 1830 excomimw 2043 exexw 2051 excom 2162 cgsex4gOLD 3529 opab0 5559 bj-modal4e 36716 |
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