Theorem 19.32dc 1569

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 1548	. . . 4 ⊢ Ⅎ𝑥 ¬ 𝜑
3	2	19.21 1475	. . 3 ⊢ (∀𝑥(¬ 𝜑 → 𝜓) ↔ (¬ 𝜑 → ∀𝑥𝜓))
4	3	a1i 9	. 2 ⊢ (DECID 𝜑 → (∀𝑥(¬ 𝜑 → 𝜓) ↔ (¬ 𝜑 → ∀𝑥𝜓)))
5	1	nfdc 1549	. . 3 ⊢ Ⅎ𝑥DECID 𝜑
6		dfordc 791	. . 3 ⊢ (DECID 𝜑 → ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓)))
7	5, 6	albid 1506	. 2 ⊢ (DECID 𝜑 → (∀𝑥(𝜑 ∨ 𝜓) ↔ ∀𝑥(¬ 𝜑 → 𝜓)))
8		dfordc 791	. 2 ⊢ (DECID 𝜑 → ((𝜑 ∨ ∀𝑥𝜓) ↔ (¬ 𝜑 → ∀𝑥𝜓)))
9	4, 7, 8	3bitr4d 209	1 ⊢ (DECID 𝜑 → (∀𝑥(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑥𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98   ∨ wo 629  DECID wdc 742  ∀wal 1241  Ⅎwnf 1349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-gen 1338  ax-ie2 1383  ax-4 1400  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-dc 743  df-tru 1246  df-fal 1249  df-nf 1350

This theorem is referenced by: (None)