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Theorem elabreximd 27809
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 2406 . . . . . 6  |-  ( y  =  A  ->  (
y  =  B  <->  A  =  B ) )
32rexbidv 2917 . . . . 5  |-  ( y  =  A  ->  ( E. x  e.  C  y  =  B  <->  E. x  e.  C  A  =  B ) )
43elabg 3196 . . . 4  |-  ( A  e.  V  ->  ( A  e.  { y  |  E. x  e.  C  y  =  B }  <->  E. x  e.  C  A  =  B ) )
51, 4syl 17 . . 3  |-  ( ph  ->  ( A  e.  {
y  |  E. x  e.  C  y  =  B }  <->  E. x  e.  C  A  =  B )
)
65biimpa 482 . 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 459 . . . . . 6  |-  ( ( ( ph  /\  x  e.  C )  /\  A  =  B )  ->  A  =  B )
10 elabreximd.5 . . . . . . 7  |-  ( (
ph  /\  x  e.  C )  ->  ps )
1110adantr 463 . . . . . 6  |-  ( ( ( ph  /\  x  e.  C )  /\  A  =  B )  ->  ps )
12 elabreximd.3 . . . . . . 7  |-  ( A  =  B  ->  ( ch 
<->  ps ) )
1312biimpar 483 . . . . . 6  |-  ( ( A  =  B  /\  ps )  ->  ch )
149, 11, 13syl2anc 659 . . . . 5  |-  ( ( ( ph  /\  x  e.  C )  /\  A  =  B )  ->  ch )
1514exp31 602 . . . 4  |-  ( ph  ->  ( x  e.  C  ->  ( A  =  B  ->  ch ) ) )
167, 8, 15rexlimd 2887 . . 3  |-  ( ph  ->  ( E. x  e.  C  A  =  B  ->  ch ) )
1716imp 427 . 2  |-  ( (
ph  /\  E. x  e.  C  A  =  B )  ->  ch )
186, 17syldan 468 1  |-  ( (
ph  /\  A  e.  { y  |  E. x  e.  C  y  =  B } )  ->  ch )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405   F/wnf 1637    e. wcel 1842   {cab 2387   E.wrex 2754
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2758  df-rex 2759  df-v 3060
This theorem is referenced by:  elabreximdv  27810  abrexss  27811  disjabrex  27860  disjabrexf  27861
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