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Theorem ssrexv 3060
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 2994 . . 3  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
21anim1d 329 . 2  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  ph )  ->  (
x  e.  B  /\  ph ) ) )
32reximdv2 2461 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 1434   E.wrex 2350    C_ wss 2974
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-rex 2355  df-in 2980  df-ss 2987
This theorem is referenced by:  iunss1  3691  moriotass  5521  tfr1onlemssrecs  5982  tfrcllemssrecs  5995  supelti  6464  lbzbi  8771  rexico  10234  alzdvds  10388  zsupcl  10476  infssuzex  10478  gcddvds  10488  dvdslegcd  10489  bj-nn0suc  10902
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