Theorem 19.33b2 1622

Description: The antecedent provides a condition implying the converse of 19.33 1477. Compare Theorem 19.33 of [Margaris] p. 90. This variation of 19.33bdc 1623 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 723	. . . . 5 ⊢ ((¬ ∃𝑥𝜑 ∨ ¬ ∃𝑥𝜓) ↔ (¬ ∃𝑥𝜓 ∨ ¬ ∃𝑥𝜑))
2		alnex 1492	. . . . . 6 ⊢ (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓)
3		alnex 1492	. . . . . 6 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
4	2, 3	orbi12i 759	. . . . 5 ⊢ ((∀𝑥 ¬ 𝜓 ∨ ∀𝑥 ¬ 𝜑) ↔ (¬ ∃𝑥𝜓 ∨ ¬ ∃𝑥𝜑))
5	1, 4	bitr4i 186	. . . 4 ⊢ ((¬ ∃𝑥𝜑 ∨ ¬ ∃𝑥𝜓) ↔ (∀𝑥 ¬ 𝜓 ∨ ∀𝑥 ¬ 𝜑))
6		pm2.53 717	. . . . . . 7 ⊢ ((𝜓 ∨ 𝜑) → (¬ 𝜓 → 𝜑))
7	6	orcoms 725	. . . . . 6 ⊢ ((𝜑 ∨ 𝜓) → (¬ 𝜓 → 𝜑))
8	7	al2imi 1451	. . . . 5 ⊢ (∀𝑥(𝜑 ∨ 𝜓) → (∀𝑥 ¬ 𝜓 → ∀𝑥𝜑))
9		pm2.53 717	. . . . . 6 ⊢ ((𝜑 ∨ 𝜓) → (¬ 𝜑 → 𝜓))
10	9	al2imi 1451	. . . . 5 ⊢ (∀𝑥(𝜑 ∨ 𝜓) → (∀𝑥 ¬ 𝜑 → ∀𝑥𝜓))
11	8, 10	orim12d 781	. . . 4 ⊢ (∀𝑥(𝜑 ∨ 𝜓) → ((∀𝑥 ¬ 𝜓 ∨ ∀𝑥 ¬ 𝜑) → (∀𝑥𝜑 ∨ ∀𝑥𝜓)))
12	5, 11	syl5bi 151	. . 3 ⊢ (∀𝑥(𝜑 ∨ 𝜓) → ((¬ ∃𝑥𝜑 ∨ ¬ ∃𝑥𝜓) → (∀𝑥𝜑 ∨ ∀𝑥𝜓)))
13	12	com12 30	. 2 ⊢ ((¬ ∃𝑥𝜑 ∨ ¬ ∃𝑥𝜓) → (∀𝑥(𝜑 ∨ 𝜓) → (∀𝑥𝜑 ∨ ∀𝑥𝜓)))
14		19.33 1477	. 2 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ∀𝑥(𝜑 ∨ 𝜓))
15	13, 14	impbid1 141	1 ⊢ ((¬ ∃𝑥𝜑 ∨ ¬ ∃𝑥𝜓) → (∀𝑥(𝜑 ∨ 𝜓) ↔ (∀𝑥𝜑 ∨ ∀𝑥𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104   ∨ wo 703  ∀wal 1346  ∃wex 1485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-gen 1442  ax-ie2 1487

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354

This theorem is referenced by:  19.33bdc  1623