Theorem pm10.53 40705

Description: Theorem *10.53 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)

Assertion

Ref	Expression
pm10.53	⊢ (¬ ∃𝑥𝜑 → ∀𝑥(𝜑 → 𝜓))

Proof of Theorem pm10.53

Step	Hyp	Ref	Expression
1		pm2.21 123	. 2 ⊢ (¬ ∃𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜓))
2		19.38 1839	. 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓))
3	1, 2	syl 17	1 ⊢ (¬ ∃𝑥𝜑 → ∀𝑥(𝜑 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1535  ∃wex 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810

This theorem depends on definitions:  df-bi 209  df-ex 1781

This theorem is referenced by: (None)