Theorem 19.33b2 1517

Description: The antecedent provides a condition implying the converse of 19.33 1370. Compare Theorem 19.33 of [Margaris] p. 90. This variation of 19.33bdc 1518 is intuitionistically valid without a decidability condition. (Contributed by Mario Carneiro, 2-Feb-2015.)

Assertion

Ref	Expression
19.33b2	⊢ ((¬ ∃xφ ∨ ¬ ∃xψ) → (∀x(φ ∨ ψ) ↔ (∀xφ ∨ ∀xψ)))

Proof of Theorem 19.33b2

Step	Hyp	Ref	Expression
1		orcom 646	. . . . 5 ⊢ ((¬ ∃xφ ∨ ¬ ∃xψ) ↔ (¬ ∃xψ ∨ ¬ ∃xφ))
2		alnex 1385	. . . . . 6 ⊢ (∀x ¬ ψ ↔ ¬ ∃xψ)
3		alnex 1385	. . . . . 6 ⊢ (∀x ¬ φ ↔ ¬ ∃xφ)
4	2, 3	orbi12i 680	. . . . 5 ⊢ ((∀x ¬ ψ ∨ ∀x ¬ φ) ↔ (¬ ∃xψ ∨ ¬ ∃xφ))
5	1, 4	bitr4i 176	. . . 4 ⊢ ((¬ ∃xφ ∨ ¬ ∃xψ) ↔ (∀x ¬ ψ ∨ ∀x ¬ φ))
6		pm2.53 640	. . . . . . 7 ⊢ ((ψ ∨ φ) → (¬ ψ → φ))
7	6	orcoms 648	. . . . . 6 ⊢ ((φ ∨ ψ) → (¬ ψ → φ))
8	7	al2imi 1344	. . . . 5 ⊢ (∀x(φ ∨ ψ) → (∀x ¬ ψ → ∀xφ))
9		pm2.53 640	. . . . . 6 ⊢ ((φ ∨ ψ) → (¬ φ → ψ))
10	9	al2imi 1344	. . . . 5 ⊢ (∀x(φ ∨ ψ) → (∀x ¬ φ → ∀xψ))
11	8, 10	orim12d 699	. . . 4 ⊢ (∀x(φ ∨ ψ) → ((∀x ¬ ψ ∨ ∀x ¬ φ) → (∀xφ ∨ ∀xψ)))
12	5, 11	syl5bi 141	. . 3 ⊢ (∀x(φ ∨ ψ) → ((¬ ∃xφ ∨ ¬ ∃xψ) → (∀xφ ∨ ∀xψ)))
13	12	com12 27	. 2 ⊢ ((¬ ∃xφ ∨ ¬ ∃xψ) → (∀x(φ ∨ ψ) → (∀xφ ∨ ∀xψ)))
14		19.33 1370	. 2 ⊢ ((∀xφ ∨ ∀xψ) → ∀x(φ ∨ ψ))
15	13, 14	impbid1 130	1 ⊢ ((¬ ∃xφ ∨ ¬ ∃xψ) → (∀x(φ ∨ ψ) ↔ (∀xφ ∨ ∀xψ)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98   ∨ wo 628  ∀wal 1240  ∃wex 1378

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

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248

This theorem is referenced by:  19.33bdc  1518