Theorem eeanv 1920

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 1516	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1516	. 2 ⊢ Ⅎ𝑥𝜓
3	1, 2	eean 1919	1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))

Syntax hints:   ∧ wa 103   ↔ wb 104  ∃wex 1480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-nf 1449

This theorem is referenced by:  eeeanv  1921  ee4anv  1922  2eu4  2107  cgsex2g  2762  cgsex4g  2763  vtocl2  2781  spc2egv  2816  spc2gv  2817  dtruarb  4170  copsex2t  4223  copsex2g  4224  opelopabsb  4238  xpmlem  5024  fununi  5256  imain  5270  brabvv  5888  spc2ed  6201  tfrlem7  6285  ener  6745  domtr  6751  unen  6782  mapen  6812  sbthlemi10  6931  ltexprlemdisj  7547  recexprlemdisj  7571  hashfacen  10749  summodc  11324