Theorem sumeq1d 6990

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 6982	. 2 |- (A = B -> sum_k e. A C = sum_k e. B C)
3	1, 2	syl 10	1 |- (ph -> sum_k e. A C = sum_k e. B C)

Syntax hints:   -> wi 3   = wceq 956  sum_csu 6979

This theorem is referenced by:  sumeq12dv 6995  sumeq12rdv 6996  fsum1slem 7008  fsump1slem 7012  fsumcllem 7014  fsum1ps 7018  fsumsplit 7020  fsumadd 7022  fsum3 7024  fsum4 7025  csbfsumlem 7026  fsumcom 7028  fsumrev 7029  fsumrev2 7030  fsummulc1 7033  fsumconst 7038  fsumcmp 7040  fsumcmpndx2 7042  fsumabs 7043  ser1ser0 7048  bcxmas 7076  fsum0diaglem2 7257  fsum0diag 7258  efaddlem24 7361  efaddlem26 7363  efaddlem27 7364  ef1tlub 7382  ef01tlub 7386  absef01tlub 7388  fsumcnlem 7989

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-rex 1650  df-v 1812  df-un 2050  df-uni 2504  df-sum 6980

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