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Theorem supeq123d 7457
 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 4225 . . . . . . 7
54notbid 287 . . . . . 6
62, 5raleqbidv 2918 . . . . 5
73breqd 4225 . . . . . . 7
83breqd 4225 . . . . . . . 8
92, 8rexeqbidv 2919 . . . . . . 7
107, 9imbi12d 313 . . . . . 6
111, 10raleqbidv 2918 . . . . 5
126, 11anbi12d 693 . . . 4
131, 12rabeqbidv 2953 . . 3
1413unieqd 4028 . 2
15 df-sup 7448 . 2
16 df-sup 7448 . 2
1714, 15, 163eqtr4g 2495 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 360   wceq 1653  wral 2707  wrex 2708  crab 2711  cuni 4017   class class class wbr 4214  csup 7447 This theorem is referenced by:  wsuceq123  25567  wlimeq12  25572  aomclem8  27138 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-uni 4018  df-br 4215  df-sup 7448
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