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  2836  cgsex4g  2837  vtocl2  2856  spc2egv  2893  spc2gv  2894  dtruarb  4274  copsex2t  4330  copsex2g  4331  opelopabsb  4347  xpmlem  5148  fununi  5388  imain  5402  brabvv  6049  spc2ed  6377  tfrlem7  6461  ener  6929  domtr  6935  unen  6967  mapen  7003  sbthlemi10  7129  ltexprlemdisj  7789  recexprlemdisj  7813  hashfacen  11053  summodc  11889