Theorem eeanv 1951

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 1542	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1542	. 2 ⊢ Ⅎ𝑥𝜓
3	1, 2	eean 1950	1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))

Syntax hints:   ∧ wa 104   ↔ wb 105  ∃wex 1506

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-nf 1475

This theorem is referenced by:  eeeanv  1952  ee4anv  1953  2eu4  2138  cgsex2g  2799  cgsex4g  2800  vtocl2  2819  spc2egv  2854  spc2gv  2855  dtruarb  4225  copsex2t  4279  copsex2g  4280  opelopabsb  4295  xpmlem  5091  fununi  5327  imain  5341  brabvv  5972  spc2ed  6300  tfrlem7  6384  ener  6847  domtr  6853  unen  6884  mapen  6916  sbthlemi10  7041  ltexprlemdisj  7690  recexprlemdisj  7714  hashfacen  10945  summodc  11565