Theorem eeanv 2380

Description: Distribute a pair of existential quantifiers over a conjunction. Combination of 19.41v 1969 and 19.42v 1973. For a version requiring fewer axioms but with additional disjoint variable conditions, see exdistrv 1975. (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 1934	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1934	. 2 ⊢ Ⅎ𝑥𝜓
3	1, 2	eean 2379	1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))

Syntax hints:   ↔ wb 208   ∧ wa 399  ∃wex 1799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-10 2175  ax-11 2191  ax-12 2212

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-ex 1800  df-nf 1804

This theorem is referenced by:  eeeanv  2381  ee4anv  2382  ee4anvOLD  2383  ttrcltr  9671