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Theorem elabreximd 23039
Description: Class substitution in an image set. (Contributed by Thierry Arnoux, 30-Dec-2016.)
Hypotheses
Ref Expression
elabreximd.1  |-  F/ x ph
elabreximd.2  |-  F/ x ch
elabreximd.3  |-  ( A  =  B  ->  ( ch 
<->  ps ) )
elabreximd.4  |-  ( ph  ->  A  e.  V )
elabreximd.5  |-  ( (
ph  /\  x  e.  C )  ->  ps )
Assertion
Ref Expression
elabreximd  |-  ( (
ph  /\  A  e.  { y  |  E. x  e.  C  y  =  B } )  ->  ch )
Distinct variable groups:    x, y, A    y, B    y, C
Allowed substitution hints:    ph( x, y)    ps( x, y)    ch( x, y)    B( x)    C( x)    V( x, y)

Proof of Theorem elabreximd
StepHypRef Expression
1 elabreximd.4 . . . . 5  |-  ( ph  ->  A  e.  V )
2 nfv 1605 . . . . . . 7  |-  F/ x  y  =  A
3 eqeq1 2289 . . . . . . 7  |-  ( y  =  A  ->  (
y  =  B  <->  A  =  B ) )
42, 3rexbid 2562 . . . . . 6  |-  ( y  =  A  ->  ( E. x  e.  C  y  =  B  <->  E. x  e.  C  A  =  B ) )
54elabg 2915 . . . . 5  |-  ( A  e.  V  ->  ( A  e.  { y  |  E. x  e.  C  y  =  B }  <->  E. x  e.  C  A  =  B ) )
61, 5syl 15 . . . 4  |-  ( ph  ->  ( A  e.  {
y  |  E. x  e.  C  y  =  B }  <->  E. x  e.  C  A  =  B )
)
76biimpd 198 . . 3  |-  ( ph  ->  ( A  e.  {
y  |  E. x  e.  C  y  =  B }  ->  E. x  e.  C  A  =  B ) )
87imp 418 . 2  |-  ( (
ph  /\  A  e.  { y  |  E. x  e.  C  y  =  B } )  ->  E. x  e.  C  A  =  B )
9 elabreximd.1 . . . 4  |-  F/ x ph
10 elabreximd.2 . . . 4  |-  F/ x ch
11 simpr 447 . . . . . 6  |-  ( ( ( ph  /\  x  e.  C )  /\  A  =  B )  ->  A  =  B )
12 elabreximd.5 . . . . . . 7  |-  ( (
ph  /\  x  e.  C )  ->  ps )
1312adantr 451 . . . . . 6  |-  ( ( ( ph  /\  x  e.  C )  /\  A  =  B )  ->  ps )
14 elabreximd.3 . . . . . . 7  |-  ( A  =  B  ->  ( ch 
<->  ps ) )
1514biimpar 471 . . . . . 6  |-  ( ( A  =  B  /\  ps )  ->  ch )
1611, 13, 15syl2anc 642 . . . . 5  |-  ( ( ( ph  /\  x  e.  C )  /\  A  =  B )  ->  ch )
1716exp31 587 . . . 4  |-  ( ph  ->  ( x  e.  C  ->  ( A  =  B  ->  ch ) ) )
189, 10, 17rexlimd 2664 . . 3  |-  ( ph  ->  ( E. x  e.  C  A  =  B  ->  ch ) )
1918imp 418 . 2  |-  ( (
ph  /\  E. x  e.  C  A  =  B )  ->  ch )
208, 19syldan 456 1  |-  ( (
ph  /\  A  e.  { y  |  E. x  e.  C  y  =  B } )  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   F/wnf 1531    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544
This theorem is referenced by:  abrexss  23040  elabreximdv  23193  disjabrex  23359  disjabrexf  23360
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790
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