Theorem sumeq1d 8250

Description: Equality deduction for sum.

Hypothesis

Ref	Expression
sumeq1d.1	|- (ph -> A = B)

Assertion

Ref	Expression
sumeq1d	|- (ph -> sum_k e. A C = sum_k e. B C)

Proof of Theorem sumeq1d

Step	Hyp	Ref	Expression
1		sumeq1d.1	. 2 |- (ph -> A = B)
2		sumeq1 8242	. 2 |- (A = B -> sum_k e. A C = sum_k e. B C)
3	1, 2	syl 12	1 |- (ph -> sum_k e. A C = sum_k e. B C)

Syntax hints:   -> wi 3   = wceq 1298  sum_csu 8239

This theorem is referenced by:  sumeq12dv 8255  sumeq12rdv 8256  fsum1slem 8268  fsump1slem 8272  fsumcllem 8274  fsum1ps 8278  fsumsplit 8280  fsumadd 8282  fsum3 8284  fsum4 8285  csbfsumlem 8286  fsumcom 8288  fsumrev 8289  fsumrev2 8290  fsummulc1 8293  fsumconst 8298  fsumcmp 8300  fsumcmpndx2 8302  fsumabs 8303  ser1ser0i 8308  bcxmas 8336  fsum0diaglem2 8519  fsum0diag 8520  efaddlem24 8623  efaddlem26 8625  efaddlem27 8626  ef1tlubi 8644  ef01tlubi 8648  absef01tlubi 8650  fsumcnlem 9267  fsum00 15820  fsumlt 15821  fsumltisum 15824  fsumleisum 15827  isumshft2 15830  fsumlt1 15831  trirn 15834  mettrifi 15847  geomcau 15849  rrnval 16013

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-rex 2110  df-v 2294  df-un 2600  df-uni 3178  df-sum 8240

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