Theorem reeanv 2779

Description: Rearrange existential quantifiers. (Contributed by NM, 9-May-1999.)

Assertion

Ref	Expression
reeanv	⊢ (∃x ∈ A ∃y ∈ B (φ ∧ ψ) ↔ (∃x ∈ A φ ∧ ∃y ∈ B ψ))

Distinct variable groups:   φ,y   ψ,x   x,y   y,A   x,B

Allowed substitution hints:   φ(x)   ψ(y)   A(x)   B(y)

Proof of Theorem reeanv

Step	Hyp	Ref	Expression
1		nfv 1619	. 2 ⊢ Ⅎyφ
2		nfv 1619	. 2 ⊢ Ⅎxψ
3	1, 2	reean 2778	1 ⊢ (∃x ∈ A ∃y ∈ B (φ ∧ ψ) ↔ (∃x ∈ A φ ∧ ∃y ∈ B ψ))

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wrex 2616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rex 2621

This theorem is referenced by:  3reeanv  2780  2ralor  2781  ltfintr  4460  ncfinraise  4482  ncfinlower  4484  nnpw1ex  4485  tfin11  4494  nnpweq  4524  sfinltfin  4536  dfxp2  5114  xpassen  6058  peano4nc  6151  ncspw1eu  6160  sbth  6207  lectr  6212