Theorem rexcom4 1815

Description: Commutation of restricted and unrestricted existential quantifiers.

Assertion

Ref	Expression
rexcom4	|- (E.x e. A E.yph <-> E.yE.x e. A ph)

Distinct variable groups:   x,y   y,A

Proof of Theorem rexcom4

Step	Hyp	Ref	Expression
1		rexcom 1767	. 2 |- (E.y e. V E.x e. A ph <-> E.x e. A E.y e. V ph)
2		rexv 1812	. 2 |- (E.y e. V E.x e. A ph <-> E.yE.x e. A ph)
3		rexv 1812	. . 3 |- (E.y e. V ph <-> E.yph)
4	3	rexbii 1660	. 2 |- (E.x e. A E.y e. V ph <-> E.x e. A E.yph)
5	1, 2, 4	3bitr3r 182	1 |- (E.x e. A E.yph <-> E.yE.x e. A ph)

Syntax hints:   <-> wb 146  E.wex 977  E.wrex 1638  Vcvv 1802

This theorem is referenced by:  uni0b 2513  cnvuni 3290  imaco 3487  aceq5lem2 4708  infcvglem1 7156  nmcopexlem1 9866  nmcfnexlem1 9895  ntunte 10340

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-rex 1642  df-v 1803

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