Theorem 19.33b 1608

Description: The antecedent provides a condition implying the converse of 19.33 1607. Compare Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 27-Mar-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 5-Jul-2014.)

Assertion

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

Proof of Theorem 19.33b

Step	Hyp	Ref	Expression
1		ianor 474	. . 3 ⊢ (¬ (∃xφ ∧ ∃xψ) ↔ (¬ ∃xφ ∨ ¬ ∃xψ))
2		alnex 1543	. . . . . 6 ⊢ (∀x ¬ φ ↔ ¬ ∃xφ)
3		pm2.53 362	. . . . . . 7 ⊢ ((φ ∨ ψ) → (¬ φ → ψ))
4	3	al2imi 1561	. . . . . 6 ⊢ (∀x(φ ∨ ψ) → (∀x ¬ φ → ∀xψ))
5	2, 4	syl5bir 209	. . . . 5 ⊢ (∀x(φ ∨ ψ) → (¬ ∃xφ → ∀xψ))
6		olc 373	. . . . 5 ⊢ (∀xψ → (∀xφ ∨ ∀xψ))
7	5, 6	syl6com 31	. . . 4 ⊢ (¬ ∃xφ → (∀x(φ ∨ ψ) → (∀xφ ∨ ∀xψ)))
8		19.30 1604	. . . . . . 7 ⊢ (∀x(φ ∨ ψ) → (∀xφ ∨ ∃xψ))
9	8	orcomd 377	. . . . . 6 ⊢ (∀x(φ ∨ ψ) → (∃xψ ∨ ∀xφ))
10	9	ord 366	. . . . 5 ⊢ (∀x(φ ∨ ψ) → (¬ ∃xψ → ∀xφ))
11		orc 374	. . . . 5 ⊢ (∀xφ → (∀xφ ∨ ∀xψ))
12	10, 11	syl6com 31	. . . 4 ⊢ (¬ ∃xψ → (∀x(φ ∨ ψ) → (∀xφ ∨ ∀xψ)))
13	7, 12	jaoi 368	. . 3 ⊢ ((¬ ∃xφ ∨ ¬ ∃xψ) → (∀x(φ ∨ ψ) → (∀xφ ∨ ∀xψ)))
14	1, 13	sylbi 187	. 2 ⊢ (¬ (∃xφ ∧ ∃xψ) → (∀x(φ ∨ ψ) → (∀xφ ∨ ∀xψ)))
15		19.33 1607	. 2 ⊢ ((∀xφ ∨ ∀xψ) → ∀x(φ ∨ ψ))
16	14, 15	impbid1 194	1 ⊢ (¬ (∃xφ ∧ ∃xψ) → (∀x(φ ∨ ψ) ↔ (∀xφ ∨ ∀xψ)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358  ∀wal 1540  ∃wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-ex 1542

This theorem is referenced by: (None)