Theorem eeor 1683

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 1671	. . 3 ⊢ (∃𝑦(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∃𝑦𝜓))
3	2	exbii 1593	. 2 ⊢ (∃𝑥∃𝑦(𝜑 ∨ 𝜓) ↔ ∃𝑥(𝜑 ∨ ∃𝑦𝜓))
4		eeor.2	. . . 4 ⊢ Ⅎ𝑥𝜓
5	4	nfex 1625	. . 3 ⊢ Ⅎ𝑥∃𝑦𝜓
6	5	19.44 1670	. 2 ⊢ (∃𝑥(𝜑 ∨ ∃𝑦𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓))
7	3, 6	bitri 183	1 ⊢ (∃𝑥∃𝑦(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓))

Syntax hints:   ↔ wb 104   ∨ wo 698  Ⅎwnf 1448  ∃wex 1480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-nf 1449

This theorem is referenced by: (None)