Theorem r19.32 44590

Description: Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers, analogous to r19.32v 3270. (Contributed by Alexander van der Vekens, 29-Jun-2017.)

Hypothesis

Ref	Expression
r19.32.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
r19.32	⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑥 ∈ 𝐴 𝜓))

Proof of Theorem r19.32

Step	Hyp	Ref	Expression
1		r19.32.1	. . . 4 ⊢ Ⅎ𝑥𝜑
2	1	nfn 1860	. . 3 ⊢ Ⅎ𝑥 ¬ 𝜑
3	2	r19.21 3140	. 2 ⊢ (∀𝑥 ∈ 𝐴 (¬ 𝜑 → 𝜓) ↔ (¬ 𝜑 → ∀𝑥 ∈ 𝐴 𝜓))
4		df-or 845	. . 3 ⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓))
5	4	ralbii 3092	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ ∀𝑥 ∈ 𝐴 (¬ 𝜑 → 𝜓))
6		df-or 845	. 2 ⊢ ((𝜑 ∨ ∀𝑥 ∈ 𝐴 𝜓) ↔ (¬ 𝜑 → ∀𝑥 ∈ 𝐴 𝜓))
7	3, 5, 6	3bitr4i 303	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑥 ∈ 𝐴 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∨ wo 844  Ⅎwnf 1786  ∀wral 3064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-or 845  df-ex 1783  df-nf 1787  df-ral 3069

This theorem is referenced by:  2reu3  44602