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Theorem rexss 3164
Description: Restricted existential quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.)
Assertion
Ref Expression
rexss  |-  ( A 
C_  B  ->  ( E. x  e.  A  ph  <->  E. x  e.  B  ( x  e.  A  /\  ph ) ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    ph( x)

Proof of Theorem rexss
StepHypRef Expression
1 ssel 3091 . . . . 5  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
21pm4.71rd 391 . . . 4  |-  ( A 
C_  B  ->  (
x  e.  A  <->  ( x  e.  B  /\  x  e.  A ) ) )
32anbi1d 460 . . 3  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  ph )  <->  ( (
x  e.  B  /\  x  e.  A )  /\  ph ) ) )
4 anass 398 . . 3  |-  ( ( ( x  e.  B  /\  x  e.  A
)  /\  ph )  <->  ( x  e.  B  /\  (
x  e.  A  /\  ph ) ) )
53, 4syl6bb 195 . 2  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  ph )  <->  ( x  e.  B  /\  (
x  e.  A  /\  ph ) ) ) )
65rexbidv2 2440 1  |-  ( A 
C_  B  ->  ( E. x  e.  A  ph  <->  E. x  e.  B  ( x  e.  A  /\  ph ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    e. wcel 1480   E.wrex 2417    C_ wss 3071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-rex 2422  df-in 3077  df-ss 3084
This theorem is referenced by:  1idprl  7398  1idpru  7399  ltexprlemm  7408  suplocexprlemmu  7526  oddnn02np1  11577  oddge22np1  11578  evennn02n  11579  evennn2n  11580
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