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Theorem ralss 3069
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 3002 . . . . 5  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
21pm4.71rd 386 . . . 4  |-  ( A 
C_  B  ->  (
x  e.  A  <->  ( x  e.  B  /\  x  e.  A ) ) )
32imbi1d 229 . . 3  |-  ( A 
C_  B  ->  (
( x  e.  A  ->  ph )  <->  ( (
x  e.  B  /\  x  e.  A )  ->  ph ) ) )
4 impexp 259 . . 3  |-  ( ( ( x  e.  B  /\  x  e.  A
)  ->  ph )  <->  ( x  e.  B  ->  ( x  e.  A  ->  ph )
) )
53, 4syl6bb 194 . 2  |-  ( A 
C_  B  ->  (
( x  e.  A  ->  ph )  <->  ( x  e.  B  ->  ( x  e.  A  ->  ph )
) ) )
65ralbidv2 2375 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    /\ wa 102    <-> wb 103    e. wcel 1434   A.wral 2353    C_ wss 2982
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-ral 2358  df-in 2988  df-ss 2995
This theorem is referenced by: (None)
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