Theorem 19.31r 1695

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 1694	. 2 ⊢ ((𝜓 ∨ ∀𝑥𝜑) → ∀𝑥(𝜓 ∨ 𝜑))
3		orcom 729	. 2 ⊢ ((∀𝑥𝜑 ∨ 𝜓) ↔ (𝜓 ∨ ∀𝑥𝜑))
4		orcom 729	. . 3 ⊢ ((𝜑 ∨ 𝜓) ↔ (𝜓 ∨ 𝜑))
5	4	albii 1484	. 2 ⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ ∀𝑥(𝜓 ∨ 𝜑))
6	2, 3, 5	3imtr4i 201	1 ⊢ ((∀𝑥𝜑 ∨ 𝜓) → ∀𝑥(𝜑 ∨ 𝜓))

Syntax hints:   → wi 4   ∨ wo 709  ∀wal 1362  Ⅎwnf 1474

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 710  ax-5 1461  ax-gen 1463  ax-4 1524

This theorem depends on definitions:  df-bi 117  df-nf 1475

This theorem is referenced by: (None)