Theorem pm11.63 45323

Description: Theorem *11.63 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)

Assertion

Ref	Expression
pm11.63	⊢ (¬ ∃𝑥∃𝑦𝜑 → ∀𝑥∀𝑦(𝜑 → 𝜓))

Proof of Theorem pm11.63

Step	Hyp	Ref	Expression
1		2nexaln 1863	. 2 ⊢ (¬ ∃𝑥∃𝑦𝜑 ↔ ∀𝑥∀𝑦 ¬ 𝜑)
2		pm2.21 124	. . 3 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
3	2	2alimi 1845	. 2 ⊢ (∀𝑥∀𝑦 ¬ 𝜑 → ∀𝑥∀𝑦(𝜑 → 𝜓))
4	1, 3	sylbi 220	1 ⊢ (¬ ∃𝑥∃𝑦𝜑 → ∀𝑥∀𝑦(𝜑 → 𝜓))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀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-ex 1813

This theorem is used by: (None)