Theorem eeanv 1983

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 1574	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1574	. 2 ⊢ Ⅎ𝑥𝜓
3	1, 2	eean 1982	1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))

Syntax hints:   ∧ wa 104   ↔ wb 105  ∃wex 1538

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580

This theorem depends on definitions:  df-bi 117  df-nf 1507

This theorem is referenced by:  eeeanv  1984  ee4anv  1985  2eu4  2171  cgsex2g  2837  cgsex4g  2838  vtocl2  2857  spc2egv  2894  spc2gv  2895  dtruarb  4279  copsex2t  4335  copsex2g  4336  opelopabsb  4352  xpmlem  5155  fununi  5395  imain  5409  brabvv  6062  spc2ed  6393  tfrlem7  6478  ener  6948  domtr  6954  unen  6986  mapen  7027  sbthlemi10  7156  ltexprlemdisj  7816  recexprlemdisj  7840  hashfacen  11090  summodc  11934