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Theorem elabreximd 23828
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 . . . 4  |-  ( ph  ->  A  e.  V )
2 eqeq1 2386 . . . . . 6  |-  ( y  =  A  ->  (
y  =  B  <->  A  =  B ) )
32rexbidv 2663 . . . . 5  |-  ( y  =  A  ->  ( E. x  e.  C  y  =  B  <->  E. x  e.  C  A  =  B ) )
43elabg 3019 . . . 4  |-  ( A  e.  V  ->  ( A  e.  { y  |  E. x  e.  C  y  =  B }  <->  E. x  e.  C  A  =  B ) )
51, 4syl 16 . . 3  |-  ( ph  ->  ( A  e.  {
y  |  E. x  e.  C  y  =  B }  <->  E. x  e.  C  A  =  B )
)
65biimpa 471 . 2  |-  ( (
ph  /\  A  e.  { y  |  E. x  e.  C  y  =  B } )  ->  E. x  e.  C  A  =  B )
7 elabreximd.1 . . . 4  |-  F/ x ph
8 elabreximd.2 . . . 4  |-  F/ x ch
9 simpr 448 . . . . . 6  |-  ( ( ( ph  /\  x  e.  C )  /\  A  =  B )  ->  A  =  B )
10 elabreximd.5 . . . . . . 7  |-  ( (
ph  /\  x  e.  C )  ->  ps )
1110adantr 452 . . . . . 6  |-  ( ( ( ph  /\  x  e.  C )  /\  A  =  B )  ->  ps )
12 elabreximd.3 . . . . . . 7  |-  ( A  =  B  ->  ( ch 
<->  ps ) )
1312biimpar 472 . . . . . 6  |-  ( ( A  =  B  /\  ps )  ->  ch )
149, 11, 13syl2anc 643 . . . . 5  |-  ( ( ( ph  /\  x  e.  C )  /\  A  =  B )  ->  ch )
1514exp31 588 . . . 4  |-  ( ph  ->  ( x  e.  C  ->  ( A  =  B  ->  ch ) ) )
167, 8, 15rexlimd 2763 . . 3  |-  ( ph  ->  ( E. x  e.  C  A  =  B  ->  ch ) )
1716imp 419 . 2  |-  ( (
ph  /\  E. x  e.  C  A  =  B )  ->  ch )
186, 17syldan 457 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   F/wnf 1550    = wceq 1649    e. wcel 1717   {cab 2366   E.wrex 2643
This theorem is referenced by:  elabreximdv  23829  abrexss  23830  disjabrex  23861  disjabrexf  23862
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 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ral 2647  df-rex 2648  df-v 2894
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