Theorem pm11.61 45131

Description: Theorem *11.61 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)

Assertion

Ref	Expression
pm11.61	⊢ (∃𝑦∀𝑥(𝜑 → 𝜓) → ∀𝑥(𝜑 → ∃𝑦𝜓))

Distinct variable group:   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥, 𝑦)

Proof of Theorem pm11.61

Step	Hyp	Ref	Expression
1		19.12 2359	. 2 ⊢ (∃𝑦∀𝑥(𝜑 → 𝜓) → ∀𝑥∃𝑦(𝜑 → 𝜓))
2		19.37v 2026	. . . 4 ⊢ (∃𝑦(𝜑 → 𝜓) ↔ (𝜑 → ∃𝑦𝜓))
3	2	biimpi 219	. . 3 ⊢ (∃𝑦(𝜑 → 𝜓) → (𝜑 → ∃𝑦𝜓))
4	3	alimi 1840	. 2 ⊢ (∀𝑥∃𝑦(𝜑 → 𝜓) → ∀𝑥(𝜑 → ∃𝑦𝜓))
5	1, 4	syl 18	1 ⊢ (∃𝑦∀𝑥(𝜑 → 𝜓) → ∀𝑥(𝜑 → ∃𝑦𝜓))

This proof depends on syntax axioms:   → wi 4  ∀wal 1567  ∃wex 1808

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-10 2175  ax-11 2191  ax-12 2212

This proof depends on definitions:  df-bi 210  df-or 861  df-ex 1809  df-nf 1813

This theorem is used by: (None)