Theorem eeor 2333

Description: Distribute existential quantifiers. (Contributed by NM, 8-Aug-1994.) Avoid ax-10 2138. (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 1879	. . 3 ⊢ (∃𝑦(𝜑 ∨ 𝜓) ↔ (∃𝑦𝜑 ∨ ∃𝑦𝜓))
2	1	exbii 1844	. 2 ⊢ (∃𝑥∃𝑦(𝜑 ∨ 𝜓) ↔ ∃𝑥(∃𝑦𝜑 ∨ ∃𝑦𝜓))
3		19.43 1879	. . 3 ⊢ (∃𝑥(∃𝑦𝜑 ∨ ∃𝑦𝜓) ↔ (∃𝑥∃𝑦𝜑 ∨ ∃𝑥∃𝑦𝜓))
4		eeor.1	. . . . . 6 ⊢ Ⅎ𝑦𝜑
5	4	19.9 2202	. . . . 5 ⊢ (∃𝑦𝜑 ↔ 𝜑)
6	5	exbii 1844	. . . 4 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑥𝜑)
7		excom 2159	. . . . 5 ⊢ (∃𝑥∃𝑦𝜓 ↔ ∃𝑦∃𝑥𝜓)
8		eeor.2	. . . . . . 7 ⊢ Ⅎ𝑥𝜓
9	8	19.9 2202	. . . . . 6 ⊢ (∃𝑥𝜓 ↔ 𝜓)
10	9	exbii 1844	. . . . 5 ⊢ (∃𝑦∃𝑥𝜓 ↔ ∃𝑦𝜓)
11	7, 10	bitri 275	. . . 4 ⊢ (∃𝑥∃𝑦𝜓 ↔ ∃𝑦𝜓)
12	6, 11	orbi12i 914	. . 3 ⊢ ((∃𝑥∃𝑦𝜑 ∨ ∃𝑥∃𝑦𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓))
13	3, 12	bitri 275	. 2 ⊢ (∃𝑥(∃𝑦𝜑 ∨ ∃𝑦𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓))
14	2, 13	bitri 275	1 ⊢ (∃𝑥∃𝑦(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓))

Syntax hints:   ↔ wb 206   ∨ wo 847  ∃wex 1775  Ⅎwnf 1779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-11 2154  ax-12 2174

This theorem depends on definitions:  df-bi 207  df-or 848  df-ex 1776  df-nf 1780

This theorem is referenced by: (None)