Theorem 19.31r 1568

Description: One direction of Theorem 19.31 of [Margaris] p. 90. The converse holds in classical logic, but not intuitionistic logic. (Contributed by Jim Kingdon, 28-Jul-2018.)

Hypothesis

Ref	Expression
19.31r.1	⊢ Ⅎxψ

Assertion

Ref	Expression
19.31r	⊢ ((∀xφ ∨ ψ) → ∀x(φ ∨ ψ))

Proof of Theorem 19.31r

Step	Hyp	Ref	Expression
1		19.31r.1	. . 3 ⊢ Ⅎxψ
2	1	19.32r 1567	. 2 ⊢ ((ψ ∨ ∀xφ) → ∀x(ψ ∨ φ))
3		orcom 646	. 2 ⊢ ((∀xφ ∨ ψ) ↔ (ψ ∨ ∀xφ))
4		orcom 646	. . 3 ⊢ ((φ ∨ ψ) ↔ (ψ ∨ φ))
5	4	albii 1356	. 2 ⊢ (∀x(φ ∨ ψ) ↔ ∀x(ψ ∨ φ))
6	2, 3, 5	3imtr4i 190	1 ⊢ ((∀xφ ∨ ψ) → ∀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: (None)