Theorem sumeq1i 6987

Description: Equality inference for sum.

Hypothesis

Ref	Expression
sumeq1i.1	|- A = B

Assertion

Ref	Expression
sumeq1i	|- sum_k e. A C = sum_k e. B C

Proof of Theorem sumeq1i

Step	Hyp	Ref	Expression
1		sumeq1i.1	. 2 |- A = B
2		sumeq1 6982	. 2 |- (A = B -> sum_k e. A C = sum_k e. B C)
3	1, 2	ax-mp 7	1 |- sum_k e. A C = sum_k e. B C

Syntax hints:   = wceq 956  sum_csu 6979

This theorem is referenced by:  sumeq12i 6989  binomlem1 7066  binomlem4 7069  binomlem6 7071  isum1clim 7197  isumnn0nn 7207  isum0split 7217  fnsmnt 7226  geoisum 7242  geoisum1 7244  geoisum1c 7245  efclt 7312  efcvg 7314  efcvgfsum 7315  reefcl 7317  efcj 7336  efaddlem6 7343  efsep 7396  ef4p 7399  ipval2 8357

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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