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Theorem 2exanali 37609
 Description: Theorem *11.521 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
Assertion
Ref Expression
2exanali (¬ ∃𝑥𝑦(𝜑 ∧ ¬ 𝜓) ↔ ∀𝑥𝑦(𝜑𝜓))

Proof of Theorem 2exanali
StepHypRef Expression
1 2nalexn 1745 . . 3 (¬ ∀𝑥𝑦(𝜑𝜓) ↔ ∃𝑥𝑦 ¬ (𝜑𝜓))
21con1bii 345 . 2 (¬ ∃𝑥𝑦 ¬ (𝜑𝜓) ↔ ∀𝑥𝑦(𝜑𝜓))
3 annim 440 . . 3 ((𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑𝜓))
432exbii 1765 . 2 (∃𝑥𝑦(𝜑 ∧ ¬ 𝜓) ↔ ∃𝑥𝑦 ¬ (𝜑𝜓))
52, 4xchnxbir 322 1 (¬ ∃𝑥𝑦(𝜑 ∧ ¬ 𝜓) ↔ ∀𝑥𝑦(𝜑𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473  ∃wex 1695 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696 This theorem is referenced by: (None)
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