Theorem eeanv 1992

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 1581	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1581	. 2 ⊢ Ⅎ𝑥𝜓
3	1, 2	eean 1991	1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))

This proof depends on syntax axioms:   ∧ wa 104   ↔ wb 105  ∃wex 1545

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587

This proof depends on definitions:  df-bi 117  df-nf 1514

This theorem is used by:  eeeanv  1993  ee4anv  1994  2eu4  2180  cgsex2g  2858  cgsex4g  2859  vtocl2  2878  spc2egv  2915  spc2gv  2916  dtruarb  4328  copsex2t  4385  copsex2g  4386  opelopabsb  4402  xpmlem  5208  fununi  5449  imain  5463  brabvv  6134  spc2ed  6469  tfrlem7  6588  ener  7066  domtr  7072  unen  7105  mapen  7146  sbthlemi10  7283  ltexprlemdisj  7973  recexprlemdisj  7997  hashfacen  11284  summodc  12150