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Theorem sumeq2w 13736
 Description: Equality theorem for sum, when the class expressions and are equal everywhere. Proved using only Extensionality. (Contributed by Mario Carneiro, 24-Jun-2014.) (Revised by Mario Carneiro, 13-Jun-2019.)
Assertion
Ref Expression
sumeq2w

Proof of Theorem sumeq2w
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 csbeq2 3405 . . . . . . . . . 10
21ifeq1d 3933 . . . . . . . . 9
32mpteq2dv 4513 . . . . . . . 8
43seqeq3d 12218 . . . . . . 7
54breq1d 4436 . . . . . 6
65anbi2d 708 . . . . 5
76rexbidv 2946 . . . 4
8 csbeq2 3405 . . . . . . . . . . 11
98mpteq2dv 4513 . . . . . . . . . 10
109seqeq3d 12218 . . . . . . . . 9
1110fveq1d 5883 . . . . . . . 8
1211eqeq2d 2443 . . . . . . 7
1312anbi2d 708 . . . . . 6
1413exbidv 1761 . . . . 5
1514rexbidv 2946 . . . 4
167, 15orbi12d 714 . . 3
1716iotabidv 5586 . 2
18 df-sum 13731 . 2
19 df-sum 13731 . 2
2017, 18, 193eqtr4g 2495 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wo 369   wa 370  wal 1435   wceq 1437  wex 1659   wcel 1870  wrex 2783  csb 3401   wss 3442  cif 3915   class class class wbr 4426   cmpt 4484  cio 5563  wf1o 5600  cfv 5601  (class class class)co 6305  cc0 9538  c1 9539   caddc 9541  cn 10609  cz 10937  cuz 11159  cfz 11782   cseq 12210   cli 13526  csu 13730 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 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407 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 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-xp 4860  df-cnv 4862  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-iota 5565  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-seq 12211  df-sum 13731 This theorem is referenced by: (None)
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