Theorem eeanv 2381

Description: Distribute a pair of existential quantifiers over a conjunction. Combination of 19.41v 1979 and 19.42v 1983. For a version requiring fewer axioms but with additional disjoint variable conditions, see exdistrv 1985. (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 1944	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1944	. 2 ⊢ Ⅎ𝑥𝜓
3	1, 2	eean 2380	1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))

Syntax hints:   ↔ wb 209   ∧ wa 400  ∃wex 1809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-10 2176  ax-11 2192  ax-12 2213

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1810  df-nf 1814

This theorem is referenced by:  eeeanv  2382  ee4anv  2383  ee4anvOLD  2384  ttrcltr  9681