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Theorem ralss 3252
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 3187 . . . . 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 2578 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 1696   A.wral 2556    C_ wss 3165
This theorem is referenced by:  acsfn  13577  acsfn1  13579  acsfn2  13581  acsfn1p  27610
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-ral 2561  df-in 3172  df-ss 3179
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