Theorem eeanv 1992

Description: Rearrange existential quantifiers. (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 1581	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1581	. 2 ⊢ Ⅎ𝑥𝜓
3	1, 2	eean 1991	1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))

Syntax hints:   ∧ wa 104   ↔ wb 105  ∃wex 1545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587

This theorem depends on definitions:  df-bi 117  df-nf 1514

This theorem is referenced by:  eeeanv  1993  ee4anv  1994  2eu4  2180  cgsex2g  2858  cgsex4g  2859  vtocl2  2878  spc2egv  2915  spc2gv  2916  dtruarb  4323  copsex2t  4380  copsex2g  4381  opelopabsb  4397  xpmlem  5203  fununi  5444  imain  5458  brabvv  6124  spc2ed  6459  tfrlem7  6578  ener  7056  domtr  7062  unen  7095  mapen  7136  sbthlemi10  7273  ltexprlemdisj  7963  recexprlemdisj  7987  hashfacen  11262  summodc  12128