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Theorem ssrexv 3132
Description: Existential quantification restricted to a subclass. (Contributed by NM, 11-Jan-2007.)
Assertion
Ref Expression
ssrexv  |-  ( A 
C_  B  ->  ( E. x  e.  A  ph 
->  E. x  e.  B  ph ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    ph( x)

Proof of Theorem ssrexv
StepHypRef Expression
1 ssel 3061 . . 3  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
21anim1d 334 . 2  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  ph )  ->  (
x  e.  B  /\  ph ) ) )
32reximdv2 2508 1  |-  ( A 
C_  B  ->  ( E. x  e.  A  ph 
->  E. x  e.  B  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1465   E.wrex 2394    C_ wss 3041
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 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-11 1469  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-rex 2399  df-in 3047  df-ss 3054
This theorem is referenced by:  iunss1  3794  moriotass  5726  tfr1onlemssrecs  6204  tfrcllemssrecs  6217  fiss  6833  supelti  6857  ctssdclemn0  6963  ctssdc  6966  enumctlemm  6967  lbzbi  9376  rexico  10961  alzdvds  11479  zsupcl  11567  infssuzex  11569  gcddvds  11579  dvdslegcd  11580  ssrest  12278  bj-nn0suc  13089
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