Theorem 19.33b 1880

Description: The antecedent provides a condition implying the converse of 19.33 1879. (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 978	. . 3 ⊢ (¬ (∃𝑥𝜑 ∧ ∃𝑥𝜓) ↔ (¬ ∃𝑥𝜑 ∨ ¬ ∃𝑥𝜓))
2		alnex 1775	. . . . . 6 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
3		pm2.53 848	. . . . . . 7 ⊢ ((𝜑 ∨ 𝜓) → (¬ 𝜑 → 𝜓))
4	3	al2imi 1809	. . . . . 6 ⊢ (∀𝑥(𝜑 ∨ 𝜓) → (∀𝑥 ¬ 𝜑 → ∀𝑥𝜓))
5	2, 4	biimtrrid 242	. . . . 5 ⊢ (∀𝑥(𝜑 ∨ 𝜓) → (¬ ∃𝑥𝜑 → ∀𝑥𝜓))
6		olc 865	. . . . 5 ⊢ (∀𝑥𝜓 → (∀𝑥𝜑 ∨ ∀𝑥𝜓))
7	5, 6	syl6com 37	. . . 4 ⊢ (¬ ∃𝑥𝜑 → (∀𝑥(𝜑 ∨ 𝜓) → (∀𝑥𝜑 ∨ ∀𝑥𝜓)))
8		19.30 1876	. . . . . . 7 ⊢ (∀𝑥(𝜑 ∨ 𝜓) → (∀𝑥𝜑 ∨ ∃𝑥𝜓))
9	8	orcomd 868	. . . . . 6 ⊢ (∀𝑥(𝜑 ∨ 𝜓) → (∃𝑥𝜓 ∨ ∀𝑥𝜑))
10	9	ord 861	. . . . 5 ⊢ (∀𝑥(𝜑 ∨ 𝜓) → (¬ ∃𝑥𝜓 → ∀𝑥𝜑))
11		orc 864	. . . . 5 ⊢ (∀𝑥𝜑 → (∀𝑥𝜑 ∨ ∀𝑥𝜓))
12	10, 11	syl6com 37	. . . 4 ⊢ (¬ ∃𝑥𝜓 → (∀𝑥(𝜑 ∨ 𝜓) → (∀𝑥𝜑 ∨ ∀𝑥𝜓)))
13	7, 12	jaoi 854	. . 3 ⊢ ((¬ ∃𝑥𝜑 ∨ ¬ ∃𝑥𝜓) → (∀𝑥(𝜑 ∨ 𝜓) → (∀𝑥𝜑 ∨ ∀𝑥𝜓)))
14	1, 13	sylbi 216	. 2 ⊢ (¬ (∃𝑥𝜑 ∧ ∃𝑥𝜓) → (∀𝑥(𝜑 ∨ 𝜓) → (∀𝑥𝜑 ∨ ∀𝑥𝜓)))
15		19.33 1879	. 2 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ∀𝑥(𝜑 ∨ 𝜓))
16	14, 15	impbid1 224	1 ⊢ (¬ (∃𝑥𝜑 ∧ ∃𝑥𝜓) → (∀𝑥(𝜑 ∨ 𝜓) ↔ (∀𝑥𝜑 ∨ ∀𝑥𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 844  ∀wal 1531  ∃wex 1773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ex 1774

This theorem is referenced by:  kmlem16  10156