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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 1782 | . 2 ⊢ (∃𝑥∃𝑦𝜑 ↔ ¬ ∀𝑥 ¬ ∃𝑦𝜑) | |
2 | alnex 1783 | . . 3 ⊢ (∀𝑦 ¬ 𝜑 ↔ ¬ ∃𝑦𝜑) | |
3 | 2 | albii 1821 | . 2 ⊢ (∀𝑥∀𝑦 ¬ 𝜑 ↔ ∀𝑥 ¬ ∃𝑦𝜑) |
4 | 1, 3 | xchbinxr 338 | 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: 2nexaln 1831 excom 2166 cbvex2v 2354 cgsex4g 3486 opab0 5406 bj-modal4e 34162 |
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