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  4330  copsex2t  4387  copsex2g  4388  opelopabsb  4404  xpmlem  5211  fununi  5452  imain  5466  brabvv  6137  spc2ed  6473  tfrlem7  6593  ener  7072  domtr  7078  unen  7111  mapen  7152  sbthlemi10  7290  ltexprlemdisj  7979  recexprlemdisj  8003  hashfacen  11305  summodc  12174