Theorem 19.32dc 1609

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 1588	. . . 4 ⊢ Ⅎ𝑥 ¬ 𝜑
3	2	19.21 1515	. . 3 ⊢ (∀𝑥(¬ 𝜑 → 𝜓) ↔ (¬ 𝜑 → ∀𝑥𝜓))
4	3	a1i 9	. 2 ⊢ (DECID 𝜑 → (∀𝑥(¬ 𝜑 → 𝜓) ↔ (¬ 𝜑 → ∀𝑥𝜓)))
5	1	nfdc 1589	. . 3 ⊢ Ⅎ𝑥DECID 𝜑
6		dfordc 824	. . 3 ⊢ (DECID 𝜑 → ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓)))
7	5, 6	albid 1546	. 2 ⊢ (DECID 𝜑 → (∀𝑥(𝜑 ∨ 𝜓) ↔ ∀𝑥(¬ 𝜑 → 𝜓)))
8		dfordc 824	. 2 ⊢ (DECID 𝜑 → ((𝜑 ∨ ∀𝑥𝜓) ↔ (¬ 𝜑 → ∀𝑥𝜓)))
9	4, 7, 8	3bitr4d 218	1 ⊢ (DECID 𝜑 → (∀𝑥(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑥𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 103   ∨ wo 661  DECID wdc 775  ∀wal 1282  Ⅎwnf 1389

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-gen 1378  ax-ie2 1423  ax-4 1440  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-dc 776  df-tru 1287  df-fal 1290  df-nf 1390

This theorem is referenced by: (None)