Theorem eeor 1885

Description: Rearrange existential quantifiers. (Contributed by NM, 8-Aug-1994.)

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		eeor.1	. . . 4 ⊢ Ⅎyφ
2	1	19.45 1878	. . 3 ⊢ (∃y(φ ∨ ψ) ↔ (φ ∨ ∃yψ))
3	2	exbii 1582	. 2 ⊢ (∃x∃y(φ ∨ ψ) ↔ ∃x(φ ∨ ∃yψ))
4		eeor.2	. . . 4 ⊢ Ⅎxψ
5	4	nfex 1843	. . 3 ⊢ Ⅎx∃yψ
6	5	19.44 1877	. 2 ⊢ (∃x(φ ∨ ∃yψ) ↔ (∃xφ ∨ ∃yψ))
7	3, 6	bitri 240	1 ⊢ (∃x∃y(φ ∨ ψ) ↔ (∃xφ ∨ ∃yψ))

Syntax hints:   ↔ wb 176   ∨ wo 357  ∃wex 1541  Ⅎwnf 1544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-ex 1542  df-nf 1545

This theorem is referenced by: (None)