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Theorem ssrexf 27787
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 3186 . . 3  |-  F/ x  A  C_  B
4 ssel 3187 . . . 4  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
54anim1d 547 . . 3  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  ph )  ->  (
x  e.  B  /\  ph ) ) )
63, 5eximd 1762 . 2  |-  ( A 
C_  B  ->  ( E. x ( x  e.  A  /\  ph )  ->  E. x ( x  e.  B  /\  ph ) ) )
7 df-rex 2562 . 2  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
8 df-rex 2562 . 2  |-  ( E. x  e.  B  ph  <->  E. x ( x  e.  B  /\  ph )
)
96, 7, 83imtr4g 261 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 358   E.wex 1531    e. wcel 1696   F/_wnfc 2419   E.wrex 2557    C_ wss 3165
This theorem is referenced by:  stoweidlem34  27886
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-in 3172  df-ss 3179
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