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Theorem rspcimedv 2899
Description: Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
rspcimdv.1  |-  ( ph  ->  A  e.  B )
rspcimedv.2  |-  ( (
ph  /\  x  =  A )  ->  ( ch  ->  ps ) )
Assertion
Ref Expression
rspcimedv  |-  ( ph  ->  ( ch  ->  E. x  e.  B  ps )
)
Distinct variable groups:    x, A    x, B    ph, x    ch, x
Allowed substitution hint:    ps( x)

Proof of Theorem rspcimedv
StepHypRef Expression
1 rspcimdv.1 . . . 4  |-  ( ph  ->  A  e.  B )
2 rspcimedv.2 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  ( ch  ->  ps ) )
32con3d 125 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  ( -.  ps  ->  -.  ch )
)
41, 3rspcimdv 2898 . . 3  |-  ( ph  ->  ( A. x  e.  B  -.  ps  ->  -. 
ch ) )
54con2d 107 . 2  |-  ( ph  ->  ( ch  ->  -.  A. x  e.  B  -.  ps ) )
6 dfrex2 2569 . 2  |-  ( E. x  e.  B  ps  <->  -. 
A. x  e.  B  -.  ps )
75, 6syl6ibr 218 1  |-  ( ph  ->  ( ch  ->  E. x  e.  B  ps )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557
This theorem is referenced by:  rspcedv  2901  fargshiftfo  28383
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-v 2803
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