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Theorem seqeq123d 11259
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 11256 . 2  |-  ( ph  ->  seq  M (  .+  ,  F )  =  seq  N (  .+  ,  F
) )
3 seqeq123d.2 . . 3  |-  ( ph  ->  .+  =  Q )
43seqeq2d 11257 . 2  |-  ( ph  ->  seq  N (  .+  ,  F )  =  seq  N ( Q ,  F
) )
5 seqeq123d.3 . . 3  |-  ( ph  ->  F  =  G )
65seqeq3d 11258 . 2  |-  ( ph  ->  seq  N ( Q ,  F )  =  seq  N ( Q ,  G ) )
72, 4, 63eqtrd 2423 1  |-  ( ph  ->  seq  M (  .+  ,  F )  =  seq  N ( Q ,  G
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    seq cseq 11250
This theorem is referenced by:  relexp0  24908  relexpsucr  24909
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-cnv 4826  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-recs 6569  df-rdg 6604  df-seq 11251
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