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Theorem rexss 3035
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 2967 . . . . 5  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
21pm4.71rd 380 . . . 4  |-  ( A 
C_  B  ->  (
x  e.  A  <->  ( x  e.  B  /\  x  e.  A ) ) )
32anbi1d 446 . . 3  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  ph )  <->  ( (
x  e.  B  /\  x  e.  A )  /\  ph ) ) )
4 anass 387 . . 3  |-  ( ( ( x  e.  B  /\  x  e.  A
)  /\  ph )  <->  ( x  e.  B  /\  (
x  e.  A  /\  ph ) ) )
53, 4syl6bb 189 . 2  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  ph )  <->  ( x  e.  B  /\  (
x  e.  A  /\  ph ) ) ) )
65rexbidv2 2346 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 101    <-> wb 102    e. wcel 1409   E.wrex 2324    C_ wss 2945
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-rex 2329  df-in 2952  df-ss 2959
This theorem is referenced by:  1idprl  6746  1idpru  6747  ltexprlemm  6756  oddnn02np1  10192  oddge22np1  10193  evennn02n  10194  evennn2n  10195
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