Theorem ralv 2703

Description: A universal quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)

Assertion

Ref	Expression
ralv	|- ( A. x e. _V ph <-> A. x ph )

Proof of Theorem ralv

Step	Hyp	Ref	Expression
1		df-ral 2421	. 2 |- ( A. x e. _V ph <-> A. x ( x e. _V -> ph ) )
2		vex 2689	. . . 4 |- x e. _V
3	2	a1bi 242	. . 3 |- ( ph <-> ( x e. _V -> ph ) )
4	3	albii 1446	. 2 |- ( A. x ph <-> A. x ( x e. _V -> ph ) )
5	1, 4	bitr4i 186	1 |- ( A. x e. _V ph <-> A. x ph )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1329   e. wcel 1480   A.wral 2416   _Vcvv 2686

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-ral 2421  df-v 2688

This theorem is referenced by:  ralcom4  2708  viin  3872  issref  4921  frecrdg  6305