Theorem eeanv 1932

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 1528	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1528	. 2 ⊢ Ⅎ𝑥𝜓
3	1, 2	eean 1931	1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))

Syntax hints:   ∧ wa 104   ↔ wb 105  ∃wex 1492

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534

This theorem depends on definitions:  df-bi 117  df-nf 1461

This theorem is referenced by:  eeeanv  1933  ee4anv  1934  2eu4  2119  cgsex2g  2774  cgsex4g  2775  vtocl2  2793  spc2egv  2828  spc2gv  2829  dtruarb  4192  copsex2t  4246  copsex2g  4247  opelopabsb  4261  xpmlem  5050  fununi  5285  imain  5299  brabvv  5921  spc2ed  6234  tfrlem7  6318  ener  6779  domtr  6785  unen  6816  mapen  6846  sbthlemi10  6965  ltexprlemdisj  7605  recexprlemdisj  7629  hashfacen  10816  summodc  11391