Theorem esumeq2d 30099

Description: Equality deduction for extended sum. (Contributed by Thierry Arnoux, 21-Sep-2016.)

Hypotheses

Ref	Expression
esumeq2d.0	|- F/ k ph
esumeq2d.1	|- ( ph -> A. k e. A B = C )

Assertion

Ref	Expression
esumeq2d	|- ( ph -> Σ* k e. A B = Σ* k e. A C )

Proof of Theorem esumeq2d

Step	Hyp	Ref	Expression
1		esumeq2d.0	. 2 |- F/ k ph
2		eqidd 2623	. 2 |- ( ph -> A = A )
3		esumeq2d.1	. . 3 |- ( ph -> A. k e. A B = C )
4	3	r19.21bi 2932	. 2 |- ( ( ph /\ k e. A ) -> B = C )
5	1, 2, 4	esumeq12dvaf 30093	1 |- ( ph -> Σ* k e. A B = Σ* k e. A C )

Syntax hints:   -> wi 4   = wceq 1483   F/wnf 1708   A.wral 2912  Σ^*cesum 30089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-iota 5851  df-fv 5896  df-ov 6653  df-esum 30090

This theorem is referenced by:  esumeq2dv  30100  esumpad  30117  esumlef  30124  esumrnmpt2  30130  voliune  30292  omssubadd  30362  carsggect  30380  omsmeas  30385  dstrvprob  30533