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Theorem ssrexf 27651
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 3333 . . 3  |-  F/ x  A  C_  B
4 ssel 3334 . . . 4  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
54anim1d 548 . . 3  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  ph )  ->  (
x  e.  B  /\  ph ) ) )
63, 5eximd 1786 . 2  |-  ( A 
C_  B  ->  ( E. x ( x  e.  A  /\  ph )  ->  E. x ( x  e.  B  /\  ph ) ) )
7 df-rex 2703 . 2  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
8 df-rex 2703 . 2  |-  ( E. x  e.  B  ph  <->  E. x ( x  e.  B  /\  ph )
)
96, 7, 83imtr4g 262 1  |-  ( A 
C_  B  ->  ( E. x  e.  A  ph 
->  E. x  e.  B  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1550    e. wcel 1725   F/_wnfc 2558   E.wrex 2698    C_ wss 3312
This theorem is referenced by:  stoweidlem34  27750
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 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-in 3319  df-ss 3326
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