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Theorem ralss 3529
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 3461 . . . . 5  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
21pm4.71rd 635 . . . 4  |-  ( A 
C_  B  ->  (
x  e.  A  <->  ( x  e.  B  /\  x  e.  A ) ) )
32imbi1d 317 . . 3  |-  ( A 
C_  B  ->  (
( x  e.  A  ->  ph )  <->  ( (
x  e.  B  /\  x  e.  A )  ->  ph ) ) )
4 impexp 446 . . 3  |-  ( ( ( x  e.  B  /\  x  e.  A
)  ->  ph )  <->  ( x  e.  B  ->  ( x  e.  A  ->  ph )
) )
53, 4syl6bb 261 . 2  |-  ( A 
C_  B  ->  (
( x  e.  A  ->  ph )  <->  ( x  e.  B  ->  ( x  e.  A  ->  ph )
) ) )
65ralbidv2 2835 1  |-  ( A 
C_  B  ->  ( A. x  e.  A  ph  <->  A. x  e.  B  ( x  e.  A  ->  ph ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1758   A.wral 2799    C_ wss 3439
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-ral 2804  df-in 3446  df-ss 3453
This theorem is referenced by:  acsfn  14719  acsfn1  14721  acsfn2  14723  mdetunilem9  18561  acsfn1p  29724
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