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Theorem seqeq1d 10175
Description: Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
Hypothesis
Ref Expression
seqeqd.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
seqeq1d  |-  ( ph  ->  seq A (  .+  ,  F )  =  seq B (  .+  ,  F ) )

Proof of Theorem seqeq1d
StepHypRef Expression
1 seqeqd.1 . 2  |-  ( ph  ->  A  =  B )
2 seqeq1 10172 . 2  |-  ( A  =  B  ->  seq A (  .+  ,  F )  =  seq B (  .+  ,  F ) )
31, 2syl 14 1  |-  ( ph  ->  seq A (  .+  ,  F )  =  seq B (  .+  ,  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1314    seqcseq 10169
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-cnv 4515  df-dm 4517  df-rn 4518  df-res 4519  df-iota 5056  df-fv 5099  df-oprab 5744  df-mpo 5745  df-recs 6168  df-frec 6254  df-seqfrec 10170
This theorem is referenced by:  seqeq123d  10178  seq3f1olemqsum  10224  bcval5  10460  bcn2  10461  seq3shft  10561  iserex  11059  iser3shft  11066  isumsplit  11211  eftlub  11306
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