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Theorem elabreximd 27082
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 2471 . . . . . 6  |-  ( y  =  A  ->  (
y  =  B  <->  A  =  B ) )
32rexbidv 2973 . . . . 5  |-  ( y  =  A  ->  ( E. x  e.  C  y  =  B  <->  E. x  e.  C  A  =  B ) )
43elabg 3251 . . . 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 484 . 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 461 . . . . . 6  |-  ( ( ( ph  /\  x  e.  C )  /\  A  =  B )  ->  A  =  B )
10 elabreximd.5 . . . . . . 7  |-  ( (
ph  /\  x  e.  C )  ->  ps )
1110adantr 465 . . . . . 6  |-  ( ( ( ph  /\  x  e.  C )  /\  A  =  B )  ->  ps )
12 elabreximd.3 . . . . . . 7  |-  ( A  =  B  ->  ( ch 
<->  ps ) )
1312biimpar 485 . . . . . 6  |-  ( ( A  =  B  /\  ps )  ->  ch )
149, 11, 13syl2anc 661 . . . . 5  |-  ( ( ( ph  /\  x  e.  C )  /\  A  =  B )  ->  ch )
1514exp31 604 . . . 4  |-  ( ph  ->  ( x  e.  C  ->  ( A  =  B  ->  ch ) ) )
167, 8, 15rexlimd 2947 . . 3  |-  ( ph  ->  ( E. x  e.  C  A  =  B  ->  ch ) )
1716imp 429 . 2  |-  ( (
ph  /\  E. x  e.  C  A  =  B )  ->  ch )
186, 17syldan 470 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 369    = wceq 1379   F/wnf 1599    e. wcel 1767   {cab 2452   E.wrex 2815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-v 3115
This theorem is referenced by:  elabreximdv  27083  abrexss  27084  disjabrex  27116  disjabrexf  27117
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