Theorem ee4anv 2356

Description: Distribute two pairs of existential quantifiers over a conjunction. For a version requiring fewer axioms but with additional disjoint variable conditions, see 4exdistrv 1958. (Contributed by NM, 31-Jul-1995.) Remove disjoint variable conditions on 𝑦, 𝑧 and 𝑥, 𝑤. (Revised by Eric Schmidt, 26-Oct-2025.)

Assertion

Ref	Expression
ee4anv	⊢ (∃𝑥∃𝑦∃𝑧∃𝑤(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤𝜓))

Distinct variable groups:   𝜑,𝑧   𝜑,𝑤   𝜓,𝑥   𝜓,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑧,𝑤)

Proof of Theorem ee4anv

Step	Hyp	Ref	Expression
1		excom 2168	. . 3 ⊢ (∃𝑦∃𝑧∃𝑤(𝜑 ∧ 𝜓) ↔ ∃𝑧∃𝑦∃𝑤(𝜑 ∧ 𝜓))
2	1	exbii 1850	. 2 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤(𝜑 ∧ 𝜓) ↔ ∃𝑥∃𝑧∃𝑦∃𝑤(𝜑 ∧ 𝜓))
3		eeanv 2354	. . 3 ⊢ (∃𝑦∃𝑤(𝜑 ∧ 𝜓) ↔ (∃𝑦𝜑 ∧ ∃𝑤𝜓))
4	3	2exbii 1851	. 2 ⊢ (∃𝑥∃𝑧∃𝑦∃𝑤(𝜑 ∧ 𝜓) ↔ ∃𝑥∃𝑧(∃𝑦𝜑 ∧ ∃𝑤𝜓))
5		nfv 1916	. . . 4 ⊢ Ⅎ𝑧𝜑
6	5	nfex 2330	. . 3 ⊢ Ⅎ𝑧∃𝑦𝜑
7		nfv 1916	. . . 4 ⊢ Ⅎ𝑥𝜓
8	7	nfex 2330	. . 3 ⊢ Ⅎ𝑥∃𝑤𝜓
9	6, 8	eean 2353	. 2 ⊢ (∃𝑥∃𝑧(∃𝑦𝜑 ∧ ∃𝑤𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤𝜓))
10	2, 4, 9	3bitri 297	1 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤𝜓))

Syntax hints:   ↔ wb 206   ∧ wa 395  ∃wex 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-10 2147  ax-11 2163  ax-12 2185

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1782  df-nf 1786

This theorem is referenced by:  5oalem7  31748  elfuns  36129