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Theorem seqeq123d 11071
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 11068 . 2  |-  ( ph  ->  seq  M (  .+  ,  F )  =  seq  N (  .+  ,  F
) )
3 seqeq123d.2 . . 3  |-  ( ph  ->  .+  =  Q )
43seqeq2d 11069 . 2  |-  ( ph  ->  seq  N (  .+  ,  F )  =  seq  N ( Q ,  F
) )
5 seqeq123d.3 . . 3  |-  ( ph  ->  F  =  G )
65seqeq3d 11070 . 2  |-  ( ph  ->  seq  N ( Q ,  F )  =  seq  N ( Q ,  G ) )
72, 4, 63eqtrd 2332 1  |-  ( ph  ->  seq  M (  .+  ,  F )  =  seq  N ( Q ,  G
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    seq cseq 11062
This theorem is referenced by:  relexp0  24040  relexpsucr  24041
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-recs 6404  df-rdg 6439  df-seq 11063
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