Theorem pm11.61 41900

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 2325	. 2 ⊢ (∃𝑦∀𝑥(𝜑 → 𝜓) → ∀𝑥∃𝑦(𝜑 → 𝜓))
2		19.37v 1996	. . . 4 ⊢ (∃𝑦(𝜑 → 𝜓) ↔ (𝜑 → ∃𝑦𝜓))
3	2	biimpi 215	. . 3 ⊢ (∃𝑦(𝜑 → 𝜓) → (𝜑 → ∃𝑦𝜓))
4	3	alimi 1815	. 2 ⊢ (∀𝑥∃𝑦(𝜑 → 𝜓) → ∀𝑥(𝜑 → ∃𝑦𝜓))
5	1, 4	syl 17	1 ⊢ (∃𝑦∀𝑥(𝜑 → 𝜓) → ∀𝑥(𝜑 → ∃𝑦𝜓))

Syntax hints:   → wi 4  ∀wal 1537  ∃wex 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2139  ax-11 2156  ax-12 2173

This theorem depends on definitions:  df-bi 206  df-or 844  df-ex 1784  df-nf 1788

This theorem is referenced by: (None)