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Theorem elabreximdv 23945
Description: Class substitution in an image set. (Contributed by Thierry Arnoux, 30-Dec-2016.)
Hypotheses
Ref Expression
elabreximdv.1  |-  ( A  =  B  ->  ( ch 
<->  ps ) )
elabreximdv.2  |-  ( ph  ->  A  e.  V )
elabreximdv.3  |-  ( (
ph  /\  x  e.  C )  ->  ps )
Assertion
Ref Expression
elabreximdv  |-  ( (
ph  /\  A  e.  { y  |  E. x  e.  C  y  =  B } )  ->  ch )
Distinct variable groups:    x, y, A    y, B    x, C, y    ch, x    ph, x
Allowed substitution hints:    ph( y)    ps( x, y)    ch( y)    B( x)    V( x, y)

Proof of Theorem elabreximdv
StepHypRef Expression
1 nfv 1626 . 2  |-  F/ x ph
2 nfv 1626 . 2  |-  F/ x ch
3 elabreximdv.1 . 2  |-  ( A  =  B  ->  ( ch 
<->  ps ) )
4 elabreximdv.2 . 2  |-  ( ph  ->  A  e.  V )
5 elabreximdv.3 . 2  |-  ( (
ph  /\  x  e.  C )  ->  ps )
61, 2, 3, 4, 5elabreximd 23944 1  |-  ( (
ph  /\  A  e.  { y  |  E. x  e.  C  y  =  B } )  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   {cab 2390   E.wrex 2667
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-rex 2672  df-v 2918
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