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Theorem seqeq123d 11916
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 11913 . 2  |-  ( ph  ->  seq M (  .+  ,  F )  =  seq N (  .+  ,  F ) )
3 seqeq123d.2 . . 3  |-  ( ph  ->  .+  =  Q )
43seqeq2d 11914 . 2  |-  ( ph  ->  seq N (  .+  ,  F )  =  seq N ( Q ,  F ) )
5 seqeq123d.3 . . 3  |-  ( ph  ->  F  =  G )
65seqeq3d 11915 . 2  |-  ( ph  ->  seq N ( Q ,  F )  =  seq N ( Q ,  G ) )
72, 4, 63eqtrd 2496 1  |-  ( ph  ->  seq M (  .+  ,  F )  =  seq N ( Q ,  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    seqcseq 11907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-opab 4449  df-mpt 4450  df-cnv 4946  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fv 5524  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-recs 6932  df-rdg 6966  df-seq 11908
This theorem is referenced by:  sseqval  26905  relexp0  27465  relexpsucr  27466  bj-finsumval0  32889
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