Theorem pm11.12 45126

Description: Theorem *11.12 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.)

Assertion

Ref	Expression
pm11.12	⊢ (∀𝑥∀𝑦(𝜑 ∨ 𝜓) → (𝜑 ∨ ∀𝑥∀𝑦𝜓))

Distinct variable groups:   𝜑,𝑥   𝜑,𝑦

Allowed substitution hints:   𝜓(𝑥, 𝑦)

Proof of Theorem pm11.12

Step	Hyp	Ref	Expression
1		pm10.12 45109	. . 3 ⊢ (∀𝑦(𝜑 ∨ 𝜓) → (𝜑 ∨ ∀𝑦𝜓))
2	1	alimi 1844	. 2 ⊢ (∀𝑥∀𝑦(𝜑 ∨ 𝜓) → ∀𝑥(𝜑 ∨ ∀𝑦𝜓))
3		pm10.12 45109	. 2 ⊢ (∀𝑥(𝜑 ∨ ∀𝑦𝜓) → (𝜑 ∨ ∀𝑥∀𝑦𝜓))
4	2, 3	syl 18	1 ⊢ (∀𝑥∀𝑦(𝜑 ∨ 𝜓) → (𝜑 ∨ ∀𝑥∀𝑦𝜓))

This proof depends on syntax axioms:   → wi 4   ∨ wo 861  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943

This proof depends on definitions:  df-bi 210  df-or 862  df-ex 1813

This theorem is used by: (None)