Theorem pm11.12 40700

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 40683	. . 3 ⊢ (∀𝑦(𝜑 ∨ 𝜓) → (𝜑 ∨ ∀𝑦𝜓))
2	1	alimi 1808	. 2 ⊢ (∀𝑥∀𝑦(𝜑 ∨ 𝜓) → ∀𝑥(𝜑 ∨ ∀𝑦𝜓))
3		pm10.12 40683	. 2 ⊢ (∀𝑥(𝜑 ∨ ∀𝑦𝜓) → (𝜑 ∨ ∀𝑥∀𝑦𝜓))
4	2, 3	syl 17	1 ⊢ (∀𝑥∀𝑦(𝜑 ∨ 𝜓) → (𝜑 ∨ ∀𝑥∀𝑦𝜓))

Syntax hints:   → wi 4   ∨ wo 843  ∀wal 1531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907

This theorem depends on definitions:  df-bi 209  df-or 844  df-ex 1777

This theorem is referenced by: (None)