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Theorem rexss 3077
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 3008 . . . . 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 ) ) )
32anbi1d 453 . . 3  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  ph )  <->  ( (
x  e.  B  /\  x  e.  A )  /\  ph ) ) )
4 anass 393 . . 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 ) ) ) )
65rexbidv2 2379 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 102    <-> wb 103    e. wcel 1436   E.wrex 2356    C_ wss 2988
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-11 1440  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-rex 2361  df-in 2994  df-ss 3001
This theorem is referenced by:  1idprl  7096  1idpru  7097  ltexprlemm  7106  oddnn02np1  10805  oddge22np1  10806  evennn02n  10807  evennn2n  10808
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