Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ssrexf Structured version   Unicode version

Theorem ssrexf 29580
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 3337 . . 3  |-  F/ x  A  C_  B
4 ssel 3338 . . . 4  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
54anim1d 559 . . 3  |-  ( A 
C_  B  ->  (
( x  e.  A  /\  ph )  ->  (
x  e.  B  /\  ph ) ) )
63, 5eximd 1815 . 2  |-  ( A 
C_  B  ->  ( E. x ( x  e.  A  /\  ph )  ->  E. x ( x  e.  B  /\  ph ) ) )
7 df-rex 2711 . 2  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
8 df-rex 2711 . 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 369   E.wex 1589    e. wcel 1755   F/_wnfc 2556   E.wrex 2706    C_ wss 3316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1590  df-nf 1593  df-sb 1700  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ral 2710  df-rex 2711  df-in 3323  df-ss 3330
This theorem is referenced by:  stoweidlem34  29675
  Copyright terms: Public domain W3C validator