Theorem 19.33-2 45133

Description: Theorem *11.421 in [WhiteheadRussell] p. 163. Theorem 19.33 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)

Assertion

Ref	Expression
19.33-2	⊢ ((∀𝑥∀𝑦𝜑 ∨ ∀𝑥∀𝑦𝜓) → ∀𝑥∀𝑦(𝜑 ∨ 𝜓))

Proof of Theorem 19.33-2

Step	Hyp	Ref	Expression
1		orc 881	. . 3 ⊢ (𝜑 → (𝜑 ∨ 𝜓))
2	1	2alimi 1845	. 2 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑥∀𝑦(𝜑 ∨ 𝜓))
3		olc 882	. . 3 ⊢ (𝜓 → (𝜑 ∨ 𝜓))
4	3	2alimi 1845	. 2 ⊢ (∀𝑥∀𝑦𝜓 → ∀𝑥∀𝑦(𝜑 ∨ 𝜓))
5	2, 4	jaoi 871	1 ⊢ ((∀𝑥∀𝑦𝜑 ∨ ∀𝑥∀𝑦𝜓) → ∀𝑥∀𝑦(𝜑 ∨ 𝜓))

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

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

This theorem is used by: (None)