Theorem 19.33 1917

Description: Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)

Assertion

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

Proof of Theorem 19.33

Step	Hyp	Ref	Expression
1		orc 881	. . 3 ⊢ (𝜑 → (𝜑 ∨ 𝜓))
2	1	alimi 1844	. 2 ⊢ (∀𝑥𝜑 → ∀𝑥(𝜑 ∨ 𝜓))
3		olc 882	. . 3 ⊢ (𝜓 → (𝜑 ∨ 𝜓))
4	3	alimi 1844	. 2 ⊢ (∀𝑥𝜓 → ∀𝑥(𝜑 ∨ 𝜓))
5	2, 4	jaoi 871	1 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ∀𝑥(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:  19.33b  1918  bj-nnfor  37491  bj-nnford  37492