Theorem eeanv 2381

Description: Distribute a pair of existential quantifiers over a conjunction. Combination of 19.41v 1983 and 19.42v 1987. For a version requiring fewer axioms but with additional disjoint variable conditions, see exdistrv 1989. (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 1948	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1948	. 2 ⊢ Ⅎ𝑥𝜓
3	1, 2	eean 2380	1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401  ∃wex 1811

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-10 2179  ax-11 2196  ax-12 2215

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1812  df-nf 1816

This theorem is used by:  eeeanv  2382  ee4anv  2383  ee4anvOLD  2384  ttrcltr  9727