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Theorem supeq123d 6878
 Description: Equality deduction for supremum. (Contributed by Stefan O'Rear, 20-Jan-2015.)
Hypotheses
Ref Expression
supeq123d.a
supeq123d.b
supeq123d.c
Assertion
Ref Expression
supeq123d

Proof of Theorem supeq123d
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 supeq123d.b . . . 4
2 supeq123d.a . . . . . 6
3 supeq123d.c . . . . . . . 8
43breqd 3940 . . . . . . 7
54notbid 656 . . . . . 6
62, 5raleqbidv 2638 . . . . 5
73breqd 3940 . . . . . . 7
83breqd 3940 . . . . . . . 8
92, 8rexeqbidv 2639 . . . . . . 7
107, 9imbi12d 233 . . . . . 6
111, 10raleqbidv 2638 . . . . 5
126, 11anbi12d 464 . . . 4
131, 12rabeqbidv 2681 . . 3
1413unieqd 3747 . 2
15 df-sup 6871 . 2
16 df-sup 6871 . 2
1714, 15, 163eqtr4g 2197 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 103   wceq 1331  wral 2416  wrex 2417  crab 2420  cuni 3736   class class class wbr 3929  csup 6869 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-uni 3737  df-br 3930  df-sup 6871 This theorem is referenced by:  infeq123d  6903
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