Theorem seqeq1d 10705

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 10702	. 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 1395   seqcseq 10699

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-mpt 4150  df-cnv 4731  df-dm 4733  df-rn 4734  df-res 4735  df-iota 5284  df-fv 5332  df-oprab 6017  df-mpo 6018  df-recs 6466  df-frec 6552  df-seqfrec 10700

This theorem is referenced by:  seqeq123d  10708  seq3f1olemqsum  10765  seqf1oglem2  10772  bcval5  11015  bcn2  11016  seq3shft  11389  iserex  11890  iser3shft  11897  isumsplit  12042  ntrivcvgap  12099  eftlub  12241  gsumfzval  13464  gsumval2  13470  mulgnndir  13728