Theorem 19.32dc 1657

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

Hypothesis

Ref	Expression
19.32dc.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
19.32dc	⊢ (DECID 𝜑 → (∀𝑥(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑥𝜓)))

Proof of Theorem 19.32dc

Step	Hyp	Ref	Expression
1		19.32dc.1	. . . . 5 ⊢ Ⅎ𝑥𝜑
2	1	nfn 1636	. . . 4 ⊢ Ⅎ𝑥 ¬ 𝜑
3	2	19.21 1562	. . 3 ⊢ (∀𝑥(¬ 𝜑 → 𝜓) ↔ (¬ 𝜑 → ∀𝑥𝜓))
4	3	a1i 9	. 2 ⊢ (DECID 𝜑 → (∀𝑥(¬ 𝜑 → 𝜓) ↔ (¬ 𝜑 → ∀𝑥𝜓)))
5	1	nfdc 1637	. . 3 ⊢ Ⅎ𝑥DECID 𝜑
6		dfordc 877	. . 3 ⊢ (DECID 𝜑 → ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓)))
7	5, 6	albid 1594	. 2 ⊢ (DECID 𝜑 → (∀𝑥(𝜑 ∨ 𝜓) ↔ ∀𝑥(¬ 𝜑 → 𝜓)))
8		dfordc 877	. 2 ⊢ (DECID 𝜑 → ((𝜑 ∨ ∀𝑥𝜓) ↔ (¬ 𝜑 → ∀𝑥𝜓)))
9	4, 7, 8	3bitr4d 219	1 ⊢ (DECID 𝜑 → (∀𝑥(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑥𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104   ∨ wo 697  DECID wdc 819  ∀wal 1329  Ⅎwnf 1436

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-gen 1425  ax-ie2 1470  ax-4 1487  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-dc 820  df-tru 1334  df-fal 1337  df-nf 1437

This theorem is referenced by: (None)