MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rexss Structured version   Unicode version

Theorem rexss 3410
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 3342 . . . . 5  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
21pm4.71rd 617 . . . 4  |-  ( A 
C_  B  ->  (
x  e.  A  <->  ( x  e.  B  /\  x  e.  A ) ) )
32anbi1d 686 . . 3  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  ph )  <->  ( (
x  e.  B  /\  x  e.  A )  /\  ph ) ) )
4 anass 631 . . 3  |-  ( ( ( x  e.  B  /\  x  e.  A
)  /\  ph )  <->  ( x  e.  B  /\  (
x  e.  A  /\  ph ) ) )
53, 4syl6bb 253 . 2  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  ph )  <->  ( x  e.  B  /\  (
x  e.  A  /\  ph ) ) ) )
65rexbidv2 2728 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    <-> wb 177    /\ wa 359    e. wcel 1725   E.wrex 2706    C_ wss 3320
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-rex 2711  df-in 3327  df-ss 3334
  Copyright terms: Public domain W3C validator