Theorem seqeq3d 10718

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
seqeq3d	|- ( ph -> seq M ( .+ , A ) = seq M ( .+ , B ) )

Proof of Theorem seqeq3d

Step	Hyp	Ref	Expression
1		seqeqd.1	. 2 |- ( ph -> A = B )
2		seqeq3 10715	. 2 |- ( A = B -> seq M ( .+ , A ) = seq M ( .+ , B ) )
3	1, 2	syl 14	1 |- ( ph -> seq M ( .+ , A ) = seq M ( .+ , B ) )

Syntax hints:   -> wi 4   = wceq 1397   seqcseq 10710

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-cnv 4733  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-recs 6471  df-frec 6557  df-seqfrec 10711

This theorem is referenced by:  seqeq123d  10719  seq3f1olemstep  10777  seq3f1olemp  10778  seqf1oglem2  10783  seqf1og  10784  exp3val  10804  sumeq1  11933  sumeq2  11937  summodc  11962  zsumdc  11963  fsum3  11966  isumz  11968  prodeq1f  12131  prodeq2w  12135  prodeq2  12136  prodmodc  12157  zproddc  12158  fprodseq  12162  prod1dc  12165  mulgval  13727  lgsval  15752  lgsval4  15768  lgsneg  15772  lgsmod  15774