Theorem seqeq1d 10527

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

Step	Hyp	Ref	Expression
1		seqeqd.1	. 2 |- ( ph -> A = B )
2		seqeq1 10524	. 2 |- ( A = B -> seq A ( .+ , F ) = seq B ( .+ , F ) )
3	1, 2	syl 14	1 |- ( ph -> seq A ( .+ , F ) = seq B ( .+ , F ) )

Syntax hints:   -> wi 4   = wceq 1364   seqcseq 10521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-cnv 4668  df-dm 4670  df-rn 4671  df-res 4672  df-iota 5216  df-fv 5263  df-oprab 5923  df-mpo 5924  df-recs 6360  df-frec 6446  df-seqfrec 10522

This theorem is referenced by:  seqeq123d  10530  seq3f1olemqsum  10587  seqf1oglem2  10594  bcval5  10837  bcn2  10838  seq3shft  10985  iserex  11485  iser3shft  11492  isumsplit  11637  ntrivcvgap  11694  eftlub  11836  gsumfzval  12977  gsumval2  12983  mulgnndir  13224