Theorem 19.32r 1567

Description: One direction of Theorem 19.32 of [Margaris] p. 90. The converse holds if φ is decidable, as seen at 19.32dc 1566. (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 632	. . 3 ⊢ (φ → (φ ∨ ψ))
3	1, 2	alrimi 1412	. 2 ⊢ (φ → ∀x(φ ∨ ψ))
4		olc 631	. . 3 ⊢ (ψ → (φ ∨ ψ))
5	4	alimi 1341	. 2 ⊢ (∀xψ → ∀x(φ ∨ ψ))
6	3, 5	jaoi 635	1 ⊢ ((φ ∨ ∀xψ) → ∀x(φ ∨ ψ))

Syntax hints:   → wi 4   ∨ wo 628  ∀wal 1240  Ⅎwnf 1346

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-gen 1335  ax-4 1397

This theorem depends on definitions:  df-bi 110  df-nf 1347

This theorem is referenced by:  19.31r  1568