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  2815  spc2egv  2850  spc2gv  2851  dtruarb  4220  copsex2t  4274  copsex2g  4275  opelopabsb  4290  xpmlem  5086  fununi  5322  imain  5336  brabvv  5964  spc2ed  6286  tfrlem7  6370  ener  6833  domtr  6839  unen  6870  mapen  6902  sbthlemi10  7025  ltexprlemdisj  7666  recexprlemdisj  7690  hashfacen  10907  summodc  11526