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Theorem seqeq123d 10195
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 10192 . 2  |-  ( ph  ->  seq M (  .+  ,  F )  =  seq N (  .+  ,  F ) )
3 seqeq123d.2 . . 3  |-  ( ph  ->  .+  =  Q )
43seqeq2d 10193 . 2  |-  ( ph  ->  seq N (  .+  ,  F )  =  seq N ( Q ,  F ) )
5 seqeq123d.3 . . 3  |-  ( ph  ->  F  =  G )
65seqeq3d 10194 . 2  |-  ( ph  ->  seq N ( Q ,  F )  =  seq N ( Q ,  G ) )
72, 4, 63eqtrd 2154 1  |-  ( ph  ->  seq M (  .+  ,  F )  =  seq N ( Q ,  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316    seqcseq 10186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-cnv 4517  df-dm 4519  df-rn 4520  df-res 4521  df-iota 5058  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-recs 6170  df-frec 6256  df-seqfrec 10187
This theorem is referenced by: (None)
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