Theorem eeanv 1848

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 1461	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1461	. 2 ⊢ Ⅎ𝑥𝜓
3	1, 2	eean 1847	1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))

Syntax hints:   ∧ wa 102   ↔ wb 103  ∃wex 1421

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-nf 1390

This theorem is referenced by:  eeeanv  1849  ee4anv  1850  2eu4  2034  cgsex2g  2635  cgsex4g  2636  vtocl2  2654  spc2egv  2687  spc2gv  2688  dtruarb  3962  copsex2t  4000  copsex2g  4001  opelopabsb  4015  xpmlem  4764  fununi  4987  imain  5001  brabvv  5571  spc2ed  5874  tfrlem7  5956  ener  6282  domtr  6288  unen  6316  ltexprlemdisj  6796  recexprlemdisj  6820