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Theorem elpreq 28235
Description: Equality wihin a pair. (Contributed by Thierry Arnoux, 23-Aug-2017.)
Hypotheses
Ref Expression
elpreq.1  |-  ( ph  ->  X  e.  { A ,  B } )
elpreq.2  |-  ( ph  ->  Y  e.  { A ,  B } )
elpreq.3  |-  ( ph  ->  ( X  =  A  <-> 
Y  =  A ) )
Assertion
Ref Expression
elpreq  |-  ( ph  ->  X  =  Y )

Proof of Theorem elpreq
StepHypRef Expression
1 simpr 468 . . 3  |-  ( (
ph  /\  X  =  A )  ->  X  =  A )
2 elpreq.3 . . . 4  |-  ( ph  ->  ( X  =  A  <-> 
Y  =  A ) )
32biimpa 492 . . 3  |-  ( (
ph  /\  X  =  A )  ->  Y  =  A )
41, 3eqtr4d 2508 . 2  |-  ( (
ph  /\  X  =  A )  ->  X  =  Y )
5 elpreq.1 . . . . 5  |-  ( ph  ->  X  e.  { A ,  B } )
6 elpri 3976 . . . . 5  |-  ( X  e.  { A ,  B }  ->  ( X  =  A  \/  X  =  B ) )
75, 6syl 17 . . . 4  |-  ( ph  ->  ( X  =  A  \/  X  =  B ) )
87orcanai 927 . . 3  |-  ( (
ph  /\  -.  X  =  A )  ->  X  =  B )
9 simpl 464 . . . 4  |-  ( (
ph  /\  -.  X  =  A )  ->  ph )
102notbid 301 . . . . 5  |-  ( ph  ->  ( -.  X  =  A  <->  -.  Y  =  A ) )
1110biimpa 492 . . . 4  |-  ( (
ph  /\  -.  X  =  A )  ->  -.  Y  =  A )
12 elpreq.2 . . . . 5  |-  ( ph  ->  Y  e.  { A ,  B } )
13 elpri 3976 . . . . 5  |-  ( Y  e.  { A ,  B }  ->  ( Y  =  A  \/  Y  =  B ) )
14 pm2.53 380 . . . . 5  |-  ( ( Y  =  A  \/  Y  =  B )  ->  ( -.  Y  =  A  ->  Y  =  B ) )
1512, 13, 143syl 18 . . . 4  |-  ( ph  ->  ( -.  Y  =  A  ->  Y  =  B ) )
169, 11, 15sylc 61 . . 3  |-  ( (
ph  /\  -.  X  =  A )  ->  Y  =  B )
178, 16eqtr4d 2508 . 2  |-  ( (
ph  /\  -.  X  =  A )  ->  X  =  Y )
184, 17pm2.61dan 808 1  |-  ( ph  ->  X  =  Y )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    \/ wo 375    /\ wa 376    = wceq 1452    e. wcel 1904   {cpr 3961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-v 3033  df-un 3395  df-sn 3960  df-pr 3962
This theorem is referenced by:  indpreima  28920
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