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Theorem pm11.52 37411
Description: Theorem *11.52 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
Assertion
Ref Expression
pm11.52 (∃𝑥𝑦(𝜑𝜓) ↔ ¬ ∀𝑥𝑦(𝜑 → ¬ 𝜓))

Proof of Theorem pm11.52
StepHypRef Expression
1 df-an 384 . . 3 ((𝜑𝜓) ↔ ¬ (𝜑 → ¬ 𝜓))
212exbii 1764 . 2 (∃𝑥𝑦(𝜑𝜓) ↔ ∃𝑥𝑦 ¬ (𝜑 → ¬ 𝜓))
3 2nalexn 1744 . 2 (¬ ∀𝑥𝑦(𝜑 → ¬ 𝜓) ↔ ∃𝑥𝑦 ¬ (𝜑 → ¬ 𝜓))
42, 3bitr4i 265 1 (∃𝑥𝑦(𝜑𝜓) ↔ ¬ ∀𝑥𝑦(𝜑 → ¬ 𝜓))
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 1712  ax-4 1727
This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695
This theorem is referenced by: (None)
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