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Theorem elpreq 23952
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 448 . . 3  |-  ( (
ph  /\  X  =  A )  ->  X  =  A )
2 elpreq.3 . . . 4  |-  ( ph  ->  ( X  =  A  <-> 
Y  =  A ) )
32biimpa 471 . . 3  |-  ( (
ph  /\  X  =  A )  ->  Y  =  A )
41, 3eqtr4d 2439 . 2  |-  ( (
ph  /\  X  =  A )  ->  X  =  Y )
5 elpreq.1 . . . . 5  |-  ( ph  ->  X  e.  { A ,  B } )
6 elpri 3794 . . . . 5  |-  ( X  e.  { A ,  B }  ->  ( X  =  A  \/  X  =  B ) )
75, 6syl 16 . . . 4  |-  ( ph  ->  ( X  =  A  \/  X  =  B ) )
87orcanai 880 . . 3  |-  ( (
ph  /\  -.  X  =  A )  ->  X  =  B )
9 simpl 444 . . . 4  |-  ( (
ph  /\  -.  X  =  A )  ->  ph )
102notbid 286 . . . . 5  |-  ( ph  ->  ( -.  X  =  A  <->  -.  Y  =  A ) )
1110biimpa 471 . . . 4  |-  ( (
ph  /\  -.  X  =  A )  ->  -.  Y  =  A )
12 elpreq.2 . . . . 5  |-  ( ph  ->  Y  e.  { A ,  B } )
13 elpri 3794 . . . . 5  |-  ( Y  e.  { A ,  B }  ->  ( Y  =  A  \/  Y  =  B ) )
14 pm2.53 363 . . . . 5  |-  ( ( Y  =  A  \/  Y  =  B )  ->  ( -.  Y  =  A  ->  Y  =  B ) )
1512, 13, 143syl 19 . . . 4  |-  ( ph  ->  ( -.  Y  =  A  ->  Y  =  B ) )
169, 11, 15sylc 58 . . 3  |-  ( (
ph  /\  -.  X  =  A )  ->  Y  =  B )
178, 16eqtr4d 2439 . 2  |-  ( (
ph  /\  -.  X  =  A )  ->  X  =  Y )
184, 17pm2.61dan 767 1  |-  ( ph  ->  X  =  Y )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721   {cpr 3775
This theorem is referenced by:  indpreima  24375
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
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 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918  df-un 3285  df-sn 3780  df-pr 3781
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