Theorem ralcom4 1823

Description: Commutation of restricted and unrestricted universal quantifiers.

Assertion

Ref	Expression
ralcom4	|- (A.x e. A A.yph <-> A.yA.x e. A ph)

Distinct variable groups:   x,y   y,A

Proof of Theorem ralcom4

Step	Hyp	Ref	Expression
1		ralcom 1774	. 2 |- (A.y e. V A.x e. A ph <-> A.x e. A A.y e. V ph)
2		ralv 1820	. 2 |- (A.y e. V A.x e. A ph <-> A.yA.x e. A ph)
3		ralv 1820	. . 3 |- (A.y e. V ph <-> A.yph)
4	3	ralbii 1667	. 2 |- (A.x e. A A.y e. V ph <-> A.x e. A A.yph)
5	1, 2, 4	3bitr3r 182	1 |- (A.x e. A A.yph <-> A.yA.x e. A ph)

Syntax hints:   <-> wb 146  A.wal 954  A.wral 1645  Vcvv 1811

This theorem is referenced by:  sbcralt 1990  sbcralgf 1992  reluni 3265  funimass4 3763  kmlem12 4776  ntreq0 7708  metcn4 7971

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-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-v 1812

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