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Theorem supeq123d 7927
 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 4467 . . . . . . 7
54notbid 294 . . . . . 6
62, 5raleqbidv 3068 . . . . 5
73breqd 4467 . . . . . . 7
83breqd 4467 . . . . . . . 8
92, 8rexeqbidv 3069 . . . . . . 7
107, 9imbi12d 320 . . . . . 6
111, 10raleqbidv 3068 . . . . 5
126, 11anbi12d 710 . . . 4
131, 12rabeqbidv 3104 . . 3
1413unieqd 4261 . 2
15 df-sup 7919 . 2
16 df-sup 7919 . 2
1714, 15, 163eqtr4g 2523 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wa 369   wceq 1395  wral 2807  wrex 2808  crab 2811  cuni 4251   class class class wbr 4456  csup 7918 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-rab 2816  df-uni 4252  df-br 4457  df-sup 7919 This theorem is referenced by:  wsuceq123  29544  wlimeq12  29549  aomclem8  31169
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