Theorem eeanv 2359

Description: Distribute a pair of existential quantifiers over a conjunction. Combination of 19.41v 1957 and 19.42v 1961. For a version requiring fewer axioms but with additional disjoint variable conditions, see exdistrv 1963. (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 1922	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1922	. 2 ⊢ Ⅎ𝑥𝜓
3	1, 2	eean 2358	1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))

Syntax hints:   ↔ wb 208   ∧ wa 397  ∃wex 1787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-10 2154  ax-11 2170  ax-12 2191

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-ex 1788  df-nf 1792

This theorem is referenced by:  eeeanv  2360  ee4anv  2361  ee4anvOLD  2362  ttrcltr  9632