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Theorem resieq 5148
 Description: A restricted identity relation is equivalent to equality in its domain. (Contributed by NM, 30-Apr-2004.)
Assertion
Ref Expression
resieq

Proof of Theorem resieq
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 breq2 4208 . . . . 5
2 eqeq2 2444 . . . . 5
31, 2bibi12d 313 . . . 4
43imbi2d 308 . . 3
5 vex 2951 . . . . 5
65opres 5147 . . . 4
7 df-br 4205 . . . 4
85ideq 5017 . . . . 5
9 df-br 4205 . . . . 5
108, 9bitr3i 243 . . . 4
116, 7, 103bitr4g 280 . . 3
124, 11vtoclg 3003 . 2
1312impcom 420 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  cop 3809   class class class wbr 4204   cid 4485   cres 4872 This theorem is referenced by:  foeqcnvco  6019  f1eqcocnv  6020  dfle2  10732  pospo  14422  dirref  14672  ustref  18240  trust  18251 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-res 4882
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