Theorem 2exanali 1879

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 1847	. . 3 ⊢ (¬ ∀𝑥∀𝑦(𝜑 → 𝜓) ↔ ∃𝑥∃𝑦 ¬ (𝜑 → 𝜓))
2	1	con1bii 358	. 2 ⊢ (¬ ∃𝑥∃𝑦 ¬ (𝜑 → 𝜓) ↔ ∀𝑥∀𝑦(𝜑 → 𝜓))
3		annim 407	. . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑 → 𝜓))
4	3	2exbii 1868	. 2 ⊢ (∃𝑥∃𝑦(𝜑 ∧ ¬ 𝜓) ↔ ∃𝑥∃𝑦 ¬ (𝜑 → 𝜓))
5	2, 4	xchnxbir 335	1 ⊢ (¬ ∃𝑥∃𝑦(𝜑 ∧ ¬ 𝜓) ↔ ∀𝑥∀𝑦(𝜑 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1557  ∃wex 1798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1799

This theorem is referenced by:  dfacycgr1  35454