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Theorem elpreq 28235
 Description: Equality wihin a pair. (Contributed by Thierry Arnoux, 23-Aug-2017.)
Hypotheses
Ref Expression
elpreq.1
elpreq.2
elpreq.3
Assertion
Ref Expression
elpreq

Proof of Theorem elpreq
StepHypRef Expression
1 simpr 468 . . 3
2 elpreq.3 . . . 4
32biimpa 492 . . 3
41, 3eqtr4d 2508 . 2
5 elpreq.1 . . . . 5
6 elpri 3976 . . . . 5
75, 6syl 17 . . . 4
87orcanai 927 . . 3
9 simpl 464 . . . 4
102notbid 301 . . . . 5
1110biimpa 492 . . . 4
12 elpreq.2 . . . . 5
13 elpri 3976 . . . . 5
14 pm2.53 380 . . . . 5
1512, 13, 143syl 18 . . . 4
169, 11, 15sylc 61 . . 3
178, 16eqtr4d 2508 . 2
184, 17pm2.61dan 808 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 189   wo 375   wa 376   wceq 1452   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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