Theorem eeanv 1925

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 1521	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1521	. 2 ⊢ Ⅎ𝑥𝜓
3	1, 2	eean 1924	1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))

Syntax hints:   ∧ wa 103   ↔ wb 104  ∃wex 1485

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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-nf 1454

This theorem is referenced by:  eeeanv  1926  ee4anv  1927  2eu4  2112  cgsex2g  2766  cgsex4g  2767  vtocl2  2785  spc2egv  2820  spc2gv  2821  dtruarb  4177  copsex2t  4230  copsex2g  4231  opelopabsb  4245  xpmlem  5031  fununi  5266  imain  5280  brabvv  5899  spc2ed  6212  tfrlem7  6296  ener  6757  domtr  6763  unen  6794  mapen  6824  sbthlemi10  6943  ltexprlemdisj  7568  recexprlemdisj  7592  hashfacen  10771  summodc  11346