Theorem r19.32 39816

Description: Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers, analogous to r19.32v 3064. (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 1768	. . 3 ⊢ Ⅎ𝑥 ¬ 𝜑
3	2	r19.21 2939	. 2 ⊢ (∀𝑥 ∈ 𝐴 (¬ 𝜑 → 𝜓) ↔ (¬ 𝜑 → ∀𝑥 ∈ 𝐴 𝜓))
4		df-or 384	. . 3 ⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓))
5	4	ralbii 2963	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ ∀𝑥 ∈ 𝐴 (¬ 𝜑 → 𝜓))
6		df-or 384	. 2 ⊢ ((𝜑 ∨ ∀𝑥 ∈ 𝐴 𝜓) ↔ (¬ 𝜑 → ∀𝑥 ∈ 𝐴 𝜓))
7	3, 5, 6	3bitr4i 291	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑥 ∈ 𝐴 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382  Ⅎwnf 1699  ∀wral 2896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-or 384  df-ex 1696  df-nf 1701  df-ral 2901

This theorem is referenced by:  2reu3  39837