Theorem pm11.52 41894

Description: Theorem *11.52 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)

Assertion

Ref	Expression
pm11.52	⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ¬ ∀𝑥∀𝑦(𝜑 → ¬ 𝜓))

Proof of Theorem pm11.52

Step	Hyp	Ref	Expression
1		df-an 396	. . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜓))
2	1	2exbii 1852	. 2 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ∃𝑥∃𝑦 ¬ (𝜑 → ¬ 𝜓))
3		2nalexn 1831	. 2 ⊢ (¬ ∀𝑥∀𝑦(𝜑 → ¬ 𝜓) ↔ ∃𝑥∃𝑦 ¬ (𝜑 → ¬ 𝜓))
4	2, 3	bitr4i 277	1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ¬ ∀𝑥∀𝑦(𝜑 → ¬ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1537  ∃wex 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784

This theorem is referenced by: (None)