Theorem 19.32dc 1566

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 1545	. . . 4 ⊢ Ⅎx ¬ φ
3	2	19.21 1472	. . 3 ⊢ (∀x(¬ φ → ψ) ↔ (¬ φ → ∀xψ))
4	3	a1i 9	. 2 ⊢ (DECID φ → (∀x(¬ φ → ψ) ↔ (¬ φ → ∀xψ)))
5	1	nfdc 1546	. . 3 ⊢ ℲxDECID φ
6		dfordc 790	. . 3 ⊢ (DECID φ → ((φ ∨ ψ) ↔ (¬ φ → ψ)))
7	5, 6	albid 1503	. 2 ⊢ (DECID φ → (∀x(φ ∨ ψ) ↔ ∀x(¬ φ → ψ)))
8		dfordc 790	. 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 628  DECID wdc 741  ∀wal 1240  Ⅎwnf 1346

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 629  ax-5 1333  ax-gen 1335  ax-ie2 1380  ax-4 1397  ax-ial 1424  ax-i5r 1425

This theorem depends on definitions:  df-bi 110  df-dc 742  df-tru 1245  df-fal 1248  df-nf 1347

This theorem is referenced by: (None)