Theorem eeor 2157

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 2094	. . 3 ⊢ (∃𝑦(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∃𝑦𝜓))
3	2	exbii 1764	. 2 ⊢ (∃𝑥∃𝑦(𝜑 ∨ 𝜓) ↔ ∃𝑥(𝜑 ∨ ∃𝑦𝜓))
4		eeor.2	. . . 4 ⊢ Ⅎ𝑥𝜓
5	4	nfex 2140	. . 3 ⊢ Ⅎ𝑥∃𝑦𝜓
6	5	19.44 2093	. 2 ⊢ (∃𝑥(𝜑 ∨ ∃𝑦𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓))
7	3, 6	bitri 263	1 ⊢ (∃𝑥∃𝑦(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓))

Syntax hints:   ↔ wb 195   ∨ wo 382  ∃wex 1695  Ⅎwnf 1699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-or 384  df-ex 1696  df-nf 1701

This theorem is referenced by: (None)