Theorem 19.43 1674

Description: Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 2-Feb-2015.)

Assertion

Ref	Expression
19.43	⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))

Proof of Theorem 19.43

Step	Hyp	Ref	Expression
1		hbe1 1541	. . . 4 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)
2		hbe1 1541	. . . 4 ⊢ (∃𝑥𝜓 → ∀𝑥∃𝑥𝜓)
3	1, 2	hbor 1592	. . 3 ⊢ ((∃𝑥𝜑 ∨ ∃𝑥𝜓) → ∀𝑥(∃𝑥𝜑 ∨ ∃𝑥𝜓))
4		19.8a 1636	. . . 4 ⊢ (𝜑 → ∃𝑥𝜑)
5		19.8a 1636	. . . 4 ⊢ (𝜓 → ∃𝑥𝜓)
6	4, 5	orim12i 764	. . 3 ⊢ ((𝜑 ∨ 𝜓) → (∃𝑥𝜑 ∨ ∃𝑥𝜓))
7	3, 6	exlimih 1639	. 2 ⊢ (∃𝑥(𝜑 ∨ 𝜓) → (∃𝑥𝜑 ∨ ∃𝑥𝜓))
8		orc 717	. . . 4 ⊢ (𝜑 → (𝜑 ∨ 𝜓))
9	8	eximi 1646	. . 3 ⊢ (∃𝑥𝜑 → ∃𝑥(𝜑 ∨ 𝜓))
10		olc 716	. . . 4 ⊢ (𝜓 → (𝜑 ∨ 𝜓))
11	10	eximi 1646	. . 3 ⊢ (∃𝑥𝜓 → ∃𝑥(𝜑 ∨ 𝜓))
12	9, 11	jaoi 721	. 2 ⊢ ((∃𝑥𝜑 ∨ ∃𝑥𝜓) → ∃𝑥(𝜑 ∨ 𝜓))
13	7, 12	impbii 126	1 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))

Syntax hints:   ↔ wb 105   ∨ wo 713  ∃wex 1538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-ial 1580

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  19.44  1728  19.45  1729  19.34  1730  sborv  1937  r19.43  2689  rexun  3384  unipr  3901  uniun  3906  unopab  4162  dmun  4929  coundi  5229  coundir  5230