Theorem 19.32r 1610

Description: One direction of Theorem 19.32 of [Margaris] p. 90. The converse holds if 𝜑 is decidable, as seen at 19.32dc 1609. (Contributed by Jim Kingdon, 28-Jul-2018.)

Hypothesis

Ref	Expression
19.32r.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
19.32r	⊢ ((𝜑 ∨ ∀𝑥𝜓) → ∀𝑥(𝜑 ∨ 𝜓))

Proof of Theorem 19.32r

Step	Hyp	Ref	Expression
1		19.32r.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		orc 665	. . 3 ⊢ (𝜑 → (𝜑 ∨ 𝜓))
3	1, 2	alrimi 1455	. 2 ⊢ (𝜑 → ∀𝑥(𝜑 ∨ 𝜓))
4		olc 664	. . 3 ⊢ (𝜓 → (𝜑 ∨ 𝜓))
5	4	alimi 1384	. 2 ⊢ (∀𝑥𝜓 → ∀𝑥(𝜑 ∨ 𝜓))
6	3, 5	jaoi 668	1 ⊢ ((𝜑 ∨ ∀𝑥𝜓) → ∀𝑥(𝜑 ∨ 𝜓))

Syntax hints:   → wi 4   ∨ wo 661  ∀wal 1282  Ⅎwnf 1389

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-gen 1378  ax-4 1440

This theorem depends on definitions:  df-bi 115  df-nf 1390

This theorem is referenced by:  19.31r  1611