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Theorem seqeq123d 12101
Description: Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
Hypotheses
Ref Expression
seqeq123d.1  |-  ( ph  ->  M  =  N )
seqeq123d.2  |-  ( ph  ->  .+  =  Q )
seqeq123d.3  |-  ( ph  ->  F  =  G )
Assertion
Ref Expression
seqeq123d  |-  ( ph  ->  seq M (  .+  ,  F )  =  seq N ( Q ,  G ) )

Proof of Theorem seqeq123d
StepHypRef Expression
1 seqeq123d.1 . . 3  |-  ( ph  ->  M  =  N )
21seqeq1d 12098 . 2  |-  ( ph  ->  seq M (  .+  ,  F )  =  seq N (  .+  ,  F ) )
3 seqeq123d.2 . . 3  |-  ( ph  ->  .+  =  Q )
43seqeq2d 12099 . 2  |-  ( ph  ->  seq N (  .+  ,  F )  =  seq N ( Q ,  F ) )
5 seqeq123d.3 . . 3  |-  ( ph  ->  F  =  G )
65seqeq3d 12100 . 2  |-  ( ph  ->  seq N ( Q ,  F )  =  seq N ( Q ,  G ) )
72, 4, 63eqtrd 2499 1  |-  ( ph  ->  seq M (  .+  ,  F )  =  seq N ( Q ,  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    seqcseq 12092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-cnv 4996  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-recs 7034  df-rdg 7068  df-seq 12093
This theorem is referenced by:  relexpsucnnr  12945  sseqval  28594  bj-finsumval0  35082
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