MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ralss Unicode version

Theorem ralss 3239
Description: Restricted universal quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.)
Assertion
Ref Expression
ralss  |-  ( A 
C_  B  ->  ( A. x  e.  A  ph  <->  A. x  e.  B  ( x  e.  A  ->  ph ) ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    ph( x)

Proof of Theorem ralss
StepHypRef Expression
1 ssel 3174 . . . . 5  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
21pm4.71rd 616 . . . 4  |-  ( A 
C_  B  ->  (
x  e.  A  <->  ( x  e.  B  /\  x  e.  A ) ) )
32imbi1d 308 . . 3  |-  ( A 
C_  B  ->  (
( x  e.  A  ->  ph )  <->  ( (
x  e.  B  /\  x  e.  A )  ->  ph ) ) )
4 impexp 433 . . 3  |-  ( ( ( x  e.  B  /\  x  e.  A
)  ->  ph )  <->  ( x  e.  B  ->  ( x  e.  A  ->  ph )
) )
53, 4syl6bb 252 . 2  |-  ( A 
C_  B  ->  (
( x  e.  A  ->  ph )  <->  ( x  e.  B  ->  ( x  e.  A  ->  ph )
) ) )
65ralbidv2 2565 1  |-  ( A 
C_  B  ->  ( A. x  e.  A  ph  <->  A. x  e.  B  ( x  e.  A  ->  ph ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1684   A.wral 2543    C_ wss 3152
This theorem is referenced by:  acsfn  13561  acsfn1  13563  acsfn2  13565  acsfn1p  27507
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-ral 2548  df-in 3159  df-ss 3166
  Copyright terms: Public domain W3C validator