Theorem 19.33 1507

Description: Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
19.33	⊢ ((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ∀𝑥(𝜑 ∨ 𝜓))

Proof of Theorem 19.33

Step	Hyp	Ref	Expression
1		orc 714	. . 3 ⊢ (𝜑 → (𝜑 ∨ 𝜓))
2	1	alimi 1478	. 2 ⊢ (∀𝑥𝜑 → ∀𝑥(𝜑 ∨ 𝜓))
3		olc 713	. . 3 ⊢ (𝜓 → (𝜑 ∨ 𝜓))
4	3	alimi 1478	. 2 ⊢ (∀𝑥𝜓 → ∀𝑥(𝜑 ∨ 𝜓))
5	2, 4	jaoi 718	1 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ∀𝑥(𝜑 ∨ 𝜓))

Syntax hints:   → wi 4   ∨ wo 710  ∀wal 1371

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-gen 1472

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  19.33b2  1652