Theorem eeanv 1948

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 1539	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1539	. 2 ⊢ Ⅎ𝑥𝜓
3	1, 2	eean 1947	1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))

Syntax hints:   ∧ wa 104   ↔ wb 105  ∃wex 1503

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-nf 1472

This theorem is referenced by:  eeeanv  1949  ee4anv  1950  2eu4  2135  cgsex2g  2796  cgsex4g  2797  vtocl2  2816  spc2egv  2851  spc2gv  2852  dtruarb  4221  copsex2t  4275  copsex2g  4276  opelopabsb  4291  xpmlem  5087  fununi  5323  imain  5337  brabvv  5965  spc2ed  6288  tfrlem7  6372  ener  6835  domtr  6841  unen  6872  mapen  6904  sbthlemi10  7027  ltexprlemdisj  7668  recexprlemdisj  7692  hashfacen  10910  summodc  11529