Theorem 19.43 1676

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 1543	. . . 4 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)
2		hbe1 1543	. . . 4 ⊢ (∃𝑥𝜓 → ∀𝑥∃𝑥𝜓)
3	1, 2	hbor 1594	. . 3 ⊢ ((∃𝑥𝜑 ∨ ∃𝑥𝜓) → ∀𝑥(∃𝑥𝜑 ∨ ∃𝑥𝜓))
4		19.8a 1638	. . . 4 ⊢ (𝜑 → ∃𝑥𝜑)
5		19.8a 1638	. . . 4 ⊢ (𝜓 → ∃𝑥𝜓)
6	4, 5	orim12i 766	. . 3 ⊢ ((𝜑 ∨ 𝜓) → (∃𝑥𝜑 ∨ ∃𝑥𝜓))
7	3, 6	exlimih 1641	. 2 ⊢ (∃𝑥(𝜑 ∨ 𝜓) → (∃𝑥𝜑 ∨ ∃𝑥𝜓))
8		orc 719	. . . 4 ⊢ (𝜑 → (𝜑 ∨ 𝜓))
9	8	eximi 1648	. . 3 ⊢ (∃𝑥𝜑 → ∃𝑥(𝜑 ∨ 𝜓))
10		olc 718	. . . 4 ⊢ (𝜓 → (𝜑 ∨ 𝜓))
11	10	eximi 1648	. . 3 ⊢ (∃𝑥𝜓 → ∃𝑥(𝜑 ∨ 𝜓))
12	9, 11	jaoi 723	. 2 ⊢ ((∃𝑥𝜑 ∨ ∃𝑥𝜓) → ∃𝑥(𝜑 ∨ 𝜓))
13	7, 12	impbii 126	1 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))

Syntax hints:   ↔ wb 105   ∨ wo 715  ∃wex 1540

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 716  ax-5 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558  ax-ial 1582

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  19.44  1730  19.45  1731  19.34  1732  sborv  1939  r19.43  2691  rexun  3387  unipr  3907  uniun  3912  unopab  4168  dmun  4938  coundi  5238  coundir  5239