Theorem 2exanali 1893

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 1861	. . 3 ⊢ (¬ ∀𝑥∀𝑦(𝜑 → 𝜓) ↔ ∃𝑥∃𝑦 ¬ (𝜑 → 𝜓))
2	1	con1bii 359	. 2 ⊢ (¬ ∃𝑥∃𝑦 ¬ (𝜑 → 𝜓) ↔ ∀𝑥∀𝑦(𝜑 → 𝜓))
3		annim 409	. . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑 → 𝜓))
4	3	2exbii 1882	. 2 ⊢ (∃𝑥∃𝑦(𝜑 ∧ ¬ 𝜓) ↔ ∃𝑥∃𝑦 ¬ (𝜑 → 𝜓))
5	2, 4	xchnxbir 336	1 ⊢ (¬ ∃𝑥∃𝑦(𝜑 ∧ ¬ 𝜓) ↔ ∀𝑥∀𝑦(𝜑 → 𝜓))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813

This theorem is used by:  dfacycgr1  35657