Theorem ralv 2706

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 2422	. 2 |- ( A. x e. _V ph <-> A. x ( x e. _V -> ph ) )
2		vex 2692	. . . 4 |- x e. _V
3	2	a1bi 242	. . 3 |- ( ph <-> ( x e. _V -> ph ) )
4	3	albii 1447	. 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 1330   e. wcel 1481   A.wral 2417   _Vcvv 2689

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 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-ral 2422  df-v 2691

This theorem is referenced by:  ralcom4  2711  viin  3880  issref  4929  frecrdg  6313