Theorem ralv 2780

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 2480	. 2 |- ( A. x e. _V ph <-> A. x ( x e. _V -> ph ) )
2		vex 2766	. . . 4 |- x e. _V
3	2	a1bi 243	. . 3 |- ( ph <-> ( x e. _V -> ph ) )
4	3	albii 1484	. 2 |- ( A. x ph <-> A. x ( x e. _V -> ph ) )
5	1, 4	bitr4i 187	1 |- ( A. x e. _V ph <-> A. x ph )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1362   e. wcel 2167   A.wral 2475   _Vcvv 2763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-ral 2480  df-v 2765

This theorem is referenced by:  ralcom4  2785  viin  3976  issref  5052  frecrdg  6466