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Theorem r2ex 1683
Description: Double restricted existential quantification.
Assertion
Ref Expression
r2ex |- (E.x e. A E.y e. B ph <-> E.xE.y((x e. A /\ y e. B) /\ ph))
Distinct variable groups:   x,y   y,A
Proof of Theorem r2ex
StepHypRef Expression
1 df-rex 1642 . 2 |- (E.x e. A E.y e. B ph <-> E.x(x e. A /\ E.y e. B ph))
2 19.42v 1303 . . . 4 |- (E.y(x e. A /\ (y e. B /\ ph)) <-> (x e. A /\ E.y(y e. B /\ ph)))
3 anass 439 . . . . 5 |- (((x e. A /\ y e. B) /\ ph) <-> (x e. A /\ (y e. B /\ ph)))
43exbii 1047 . . . 4 |- (E.y((x e. A /\ y e. B) /\ ph) <-> E.y(x e. A /\ (y e. B /\ ph)))
5 df-rex 1642 . . . . 5 |- (E.y e. B ph <-> E.y(y e. B /\ ph))
65anbi2i 479 . . . 4 |- ((x e. A /\ E.y e. B ph) <-> (x e. A /\ E.y(y e. B /\ ph)))
72, 4, 63bitr4 183 . . 3 |- (E.y((x e. A /\ y e. B) /\ ph) <-> (x e. A /\ E.y e. B ph))
87exbii 1047 . 2 |- (E.xE.y((x e. A /\ y e. B) /\ ph) <-> E.x(x e. A /\ E.y e. B ph))
91, 8bitr4 176 1 |- (E.x e. A E.y e. B ph <-> E.xE.y((x e. A /\ y e. B) /\ ph))
Colors of variables: wff set class
Syntax hints:   <-> wb 146   /\ wa 223   e. wcel 955  E.wex 977  E.wrex 1638
This theorem is referenced by:  rexcom 1767  genpv 5074  axcnre 5258  pjtheu 9150  pjpj0 9170  spanun 9382  osumlem7 9501  5oalem7 9522  3oalem3 9526  bsi 10382
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 960  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-rex 1642
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