Theorem 2exanali 38587

Description: Theorem *11.521 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)

Assertion

Ref	Expression
2exanali	⊢ (¬ ∃𝑥∃𝑦(𝜑 ∧ ¬ 𝜓) ↔ ∀𝑥∀𝑦(𝜑 → 𝜓))

Proof of Theorem 2exanali

Step	Hyp	Ref	Expression
1		2nalexn 1755	. . 3 ⊢ (¬ ∀𝑥∀𝑦(𝜑 → 𝜓) ↔ ∃𝑥∃𝑦 ¬ (𝜑 → 𝜓))
2	1	con1bii 346	. 2 ⊢ (¬ ∃𝑥∃𝑦 ¬ (𝜑 → 𝜓) ↔ ∀𝑥∀𝑦(𝜑 → 𝜓))
3		annim 441	. . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑 → 𝜓))
4	3	2exbii 1775	. 2 ⊢ (∃𝑥∃𝑦(𝜑 ∧ ¬ 𝜓) ↔ ∃𝑥∃𝑦 ¬ (𝜑 → 𝜓))
5	2, 4	xchnxbir 323	1 ⊢ (¬ ∃𝑥∃𝑦(𝜑 ∧ ¬ 𝜓) ↔ ∀𝑥∀𝑦(𝜑 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1481  ∃wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by: (None)