Theorem eeanv 2341

Description: Distribute a pair of existential quantifiers over a conjunction. Combination of 19.41v 1946 and 19.42v 1950. For a version requiring fewer axioms but with additional disjoint variable conditions, see exdistrv 1952. (Contributed by NM, 26-Jul-1995.)

Assertion

Ref	Expression
eeanv	⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))

Distinct variable groups:   𝜑,𝑦   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem eeanv

Step	Hyp	Ref	Expression
1		nfv 1910	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1910	. 2 ⊢ Ⅎ𝑥𝜓
3	1, 2	eean 2340	1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))

Syntax hints:   ↔ wb 205   ∧ wa 395  ∃wex 1774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-10 2130  ax-11 2147  ax-12 2167

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-ex 1775  df-nf 1779

This theorem is referenced by:  eeeanv  2342  ee4anv  2343  ttrcltr  9733