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Theorem resieq 5126
 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 4421 . . . . 5
2 eqeq2 2435 . . . . 5
31, 2bibi12d 322 . . . 4
43imbi2d 317 . . 3
5 vex 3081 . . . . 5
65opres 5125 . . . 4
7 df-br 4418 . . . 4
85ideq 4998 . . . . 5
9 df-br 4418 . . . . 5
108, 9bitr3i 254 . . . 4
116, 7, 103bitr4g 291 . . 3
124, 11vtoclg 3136 . 2
1312impcom 431 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 187   wa 370   wceq 1437   wcel 1867  cop 3999   class class class wbr 4417   cid 4755   cres 4847 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-br 4418  df-opab 4476  df-id 4760  df-xp 4851  df-rel 4852  df-res 4857 This theorem is referenced by:  foeqcnvco  6204  f1eqcocnv  6205  dfle2  11435  pospo  16163  dirref  16425  ustref  21157  trust  21168  brfvrcld2  35927
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