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Theorem 2exanali 37407
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 1743 . . 3 (¬ ∀𝑥𝑦(𝜑𝜓) ↔ ∃𝑥𝑦 ¬ (𝜑𝜓))
21con1bii 344 . 2 (¬ ∃𝑥𝑦 ¬ (𝜑𝜓) ↔ ∀𝑥𝑦(𝜑𝜓))
3 annim 439 . . 3 ((𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑𝜓))
432exbii 1763 . 2 (∃𝑥𝑦(𝜑 ∧ ¬ 𝜓) ↔ ∃𝑥𝑦 ¬ (𝜑𝜓))
52, 4xchnxbir 321 1 (¬ ∃𝑥𝑦(𝜑 ∧ ¬ 𝜓) ↔ ∀𝑥𝑦(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  wal 1472  wex 1694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726
This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695
This theorem is referenced by: (None)
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