Theorem r19.32vdc 2619

Description: Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers, where 𝜑 is decidable. (Contributed by Jim Kingdon, 4-Jun-2018.)

Assertion

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

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem r19.32vdc

Step	Hyp	Ref	Expression
1		r19.21v 2547	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (¬ 𝜑 → 𝜓) ↔ (¬ 𝜑 → ∀𝑥 ∈ 𝐴 𝜓))
2	1	a1i 9	. 2 ⊢ (DECID 𝜑 → (∀𝑥 ∈ 𝐴 (¬ 𝜑 → 𝜓) ↔ (¬ 𝜑 → ∀𝑥 ∈ 𝐴 𝜓)))
3		dfordc 887	. . 3 ⊢ (DECID 𝜑 → ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓)))
4	3	ralbidv 2470	. 2 ⊢ (DECID 𝜑 → (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ ∀𝑥 ∈ 𝐴 (¬ 𝜑 → 𝜓)))
5		dfordc 887	. 2 ⊢ (DECID 𝜑 → ((𝜑 ∨ ∀𝑥 ∈ 𝐴 𝜓) ↔ (¬ 𝜑 → ∀𝑥 ∈ 𝐴 𝜓)))
6	2, 4, 5	3bitr4d 219	1 ⊢ (DECID 𝜑 → (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑥 ∈ 𝐴 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104   ∨ wo 703  DECID wdc 829  ∀wral 2448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-gen 1442  ax-4 1503  ax-17 1519  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-dc 830  df-nf 1454  df-ral 2453

This theorem is referenced by: (None)