Theorem seqeq3d 9929

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 9926	. 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 1290   seqcseq 9915

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2624  df-un 3006  df-in 3008  df-ss 3015  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-br 3854  df-opab 3908  df-mpt 3909  df-cnv 4462  df-dm 4464  df-rn 4465  df-res 4466  df-iota 4995  df-fv 5038  df-ov 5671  df-oprab 5672  df-mpt2 5673  df-recs 6086  df-frec 6172  df-iseq 9916  df-seq3 9917

This theorem is referenced by:  seqeq123d  9930  seq3f1olemstep  9993  seq3f1olemp  9994  exp3val  10020