Theorem eeorOLD 2336

Description: Obsolete version of eeor 2335 as of 21-Nov-2024. (Contributed by NM, 8-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

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

Assertion

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

Proof of Theorem eeorOLD

Step	Hyp	Ref	Expression
1		eeor.1	. . . 4 ⊢ Ⅎ𝑦𝜑
2	1	19.45 2238	. . 3 ⊢ (∃𝑦(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∃𝑦𝜓))
3	2	exbii 1848	. 2 ⊢ (∃𝑥∃𝑦(𝜑 ∨ 𝜓) ↔ ∃𝑥(𝜑 ∨ ∃𝑦𝜓))
4		eeor.2	. . . 4 ⊢ Ⅎ𝑥𝜓
5	4	nfex 2324	. . 3 ⊢ Ⅎ𝑥∃𝑦𝜓
6	5	19.44 2237	. 2 ⊢ (∃𝑥(𝜑 ∨ ∃𝑦𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓))
7	3, 6	bitri 275	1 ⊢ (∃𝑥∃𝑦(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓))

Syntax hints:   ↔ wb 206   ∨ wo 848  ∃wex 1779  Ⅎwnf 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-10 2141  ax-11 2157  ax-12 2177

This theorem depends on definitions:  df-bi 207  df-or 849  df-ex 1780  df-nf 1784

This theorem is referenced by: (None)