Theorem eeor 2342

Description: Distribute existential quantifiers. (Contributed by NM, 8-Aug-1994.) Avoid ax-10 2152. (Revised by GG, 21-Nov-2024.)

Hypotheses

Ref	Expression
eeor.1	⊢ Ⅎ𝑦𝜑
eeor.2	⊢ Ⅎ𝑥𝜓

Assertion

Ref	Expression
eeor	⊢ (∃𝑥∃𝑦(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓))

Proof of Theorem eeor

Step	Hyp	Ref	Expression
1		19.43 1889	. . 3 ⊢ (∃𝑦(𝜑 ∨ 𝜓) ↔ (∃𝑦𝜑 ∨ ∃𝑦𝜓))
2	1	exbii 1855	. 2 ⊢ (∃𝑥∃𝑦(𝜑 ∨ 𝜓) ↔ ∃𝑥(∃𝑦𝜑 ∨ ∃𝑦𝜓))
3		19.43 1889	. . 3 ⊢ (∃𝑥(∃𝑦𝜑 ∨ ∃𝑦𝜓) ↔ (∃𝑥∃𝑦𝜑 ∨ ∃𝑥∃𝑦𝜓))
4		eeor.1	. . . . . 6 ⊢ Ⅎ𝑦𝜑
5	4	19.9 2217	. . . . 5 ⊢ (∃𝑦𝜑 ↔ 𝜑)
6	5	exbii 1855	. . . 4 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑥𝜑)
7		excom 2173	. . . . 5 ⊢ (∃𝑥∃𝑦𝜓 ↔ ∃𝑦∃𝑥𝜓)
8		eeor.2	. . . . . . 7 ⊢ Ⅎ𝑥𝜓
9	8	19.9 2217	. . . . . 6 ⊢ (∃𝑥𝜓 ↔ 𝜓)
10	9	exbii 1855	. . . . 5 ⊢ (∃𝑦∃𝑥𝜓 ↔ ∃𝑦𝜓)
11	7, 10	bitri 276	. . . 4 ⊢ (∃𝑥∃𝑦𝜓 ↔ ∃𝑦𝜓)
12	6, 11	orbi12i 920	. . 3 ⊢ ((∃𝑥∃𝑦𝜑 ∨ ∃𝑥∃𝑦𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓))
13	3, 12	bitri 276	. 2 ⊢ (∃𝑥(∃𝑦𝜑 ∨ ∃𝑦𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓))
14	2, 13	bitri 276	1 ⊢ (∃𝑥∃𝑦(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓))

Syntax hints:   ↔ wb 207   ∨ wo 853  ∃wex 1786  Ⅎwnf 1790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-11 2168  ax-12 2189

This theorem depends on definitions:  df-bi 208  df-or 854  df-ex 1787  df-nf 1791

This theorem is referenced by: (None)