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Theorem ssrexf 3501
Description: restricted existential quantification follows from a subclass relationship. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
ssrexf.1  |-  F/_ x A
ssrexf.2  |-  F/_ x B
Assertion
Ref Expression
ssrexf  |-  ( A 
C_  B  ->  ( E. x  e.  A  ph 
->  E. x  e.  B  ph ) )

Proof of Theorem ssrexf
StepHypRef Expression
1 ssrexf.1 . . . 4  |-  F/_ x A
2 ssrexf.2 . . . 4  |-  F/_ x B
31, 2nfss 3434 . . 3  |-  F/ x  A  C_  B
4 ssel 3435 . . . 4  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
54anim1d 562 . . 3  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  ph )  ->  (
x  e.  B  /\  ph ) ) )
63, 5eximd 1906 . 2  |-  ( A 
C_  B  ->  ( E. x ( x  e.  A  /\  ph )  ->  E. x ( x  e.  B  /\  ph ) ) )
7 df-rex 2759 . 2  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
8 df-rex 2759 . 2  |-  ( E. x  e.  B  ph  <->  E. x ( x  e.  B  /\  ph )
)
96, 7, 83imtr4g 270 1  |-  ( A 
C_  B  ->  ( E. x  e.  A  ph 
->  E. x  e.  B  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367   E.wex 1633    e. wcel 1842   F/_wnfc 2550   E.wrex 2754    C_ wss 3413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2758  df-rex 2759  df-in 3420  df-ss 3427
This theorem is referenced by:  iunxdif3  27843  stoweidlem34  37165
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