Theorem pm11.63 42013

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 1832	. 2 ⊢ (¬ ∃𝑥∃𝑦𝜑 ↔ ∀𝑥∀𝑦 ¬ 𝜑)
2		pm2.21 123	. . 3 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
3	2	2alimi 1815	. 2 ⊢ (∀𝑥∀𝑦 ¬ 𝜑 → ∀𝑥∀𝑦(𝜑 → 𝜓))
4	1, 3	sylbi 216	1 ⊢ (¬ ∃𝑥∃𝑦𝜑 → ∀𝑥∀𝑦(𝜑 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1537  ∃wex 1782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812

This theorem depends on definitions:  df-bi 206  df-ex 1783

This theorem is referenced by: (None)