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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 1777 | . 2 ⊢ (∃𝑥∃𝑦𝜑 ↔ ¬ ∀𝑥 ¬ ∃𝑦𝜑) | |
2 | alnex 1778 | . . 3 ⊢ (∀𝑦 ¬ 𝜑 ↔ ¬ ∃𝑦𝜑) | |
3 | 2 | albii 1816 | . 2 ⊢ (∀𝑥∀𝑦 ¬ 𝜑 ↔ ∀𝑥 ¬ ∃𝑦𝜑) |
4 | 1, 3 | xchbinxr 337 | 1 ⊢ (∃𝑥∃𝑦𝜑 ↔ ¬ ∀𝑥∀𝑦 ¬ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 ∀wal 1531 ∃wex 1776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 |
This theorem depends on definitions: df-bi 209 df-ex 1777 |
This theorem is referenced by: 2nexaln 1826 excom 2165 cbvex2v 2361 opab0 5433 bj-modal4e 34044 |
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