Theorem 19.31r 1571

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 1570	. 2 ⊢ ((𝜓 ∨ ∀𝑥𝜑) → ∀𝑥(𝜓 ∨ 𝜑))
3		orcom 647	. 2 ⊢ ((∀𝑥𝜑 ∨ 𝜓) ↔ (𝜓 ∨ ∀𝑥𝜑))
4		orcom 647	. . 3 ⊢ ((𝜑 ∨ 𝜓) ↔ (𝜓 ∨ 𝜑))
5	4	albii 1359	. 2 ⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ ∀𝑥(𝜓 ∨ 𝜑))
6	2, 3, 5	3imtr4i 190	1 ⊢ ((∀𝑥𝜑 ∨ 𝜓) → ∀𝑥(𝜑 ∨ 𝜓))

Syntax hints:   → wi 4   ∨ wo 629  ∀wal 1241  Ⅎwnf 1349

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 630  ax-5 1336  ax-gen 1338  ax-4 1400

This theorem depends on definitions:  df-bi 110  df-nf 1350

This theorem is referenced by: (None)