Theorem seqeq1d 10670

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 10667	. 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 10664

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 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-mpt 4146  df-cnv 4726  df-dm 4728  df-rn 4729  df-res 4730  df-iota 5277  df-fv 5325  df-oprab 6004  df-mpo 6005  df-recs 6449  df-frec 6535  df-seqfrec 10665

This theorem is referenced by:  seqeq123d  10673  seq3f1olemqsum  10730  seqf1oglem2  10737  bcval5  10980  bcn2  10981  seq3shft  11344  iserex  11845  iser3shft  11852  isumsplit  11997  ntrivcvgap  12054  eftlub  12196  gsumfzval  13419  gsumval2  13425  mulgnndir  13683