Theorem pm10.53 45134

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		pm2.21 124	. 2 ⊢ (¬ ∃𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜓))
2		19.38 1872	. 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓))
3	1, 2	syl 18	1 ⊢ (¬ ∃𝑥𝜑 → ∀𝑥(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)