Theorem rexcom4 1824

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 1775	. 2 |- (E.y e. V E.x e. A ph <-> E.x e. A E.y e. V ph)
2		rexv 1821	. 2 |- (E.y e. V E.x e. A ph <-> E.yE.x e. A ph)
3		rexv 1821	. . 3 |- (E.y e. V ph <-> E.yph)
4	3	rexbii 1668	. 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 980  E.wrex 1646  Vcvv 1811

This theorem is referenced by:  uni0b 2523  cnvuni 3301  imaco 3501  aceq5lem2 4736  infcvglem1 7221  nmcopexlem1 9951  nmcfnexlem1 9980  ntunte 10439

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-rex 1650  df-v 1812

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