Theorem seqeq1d 10890

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 10887	. 2 |- ( A = B -> seq A ( .+ , F ) = seq B ( .+ , F ) )
3	1, 2	syl 14	1 |- ( ph -> seq A ( .+ , F ) = seq B ( .+ , F ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   seqcseq 10884

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-cnv 4782  df-dm 4784  df-rn 4785  df-res 4786  df-iota 5337  df-fv 5385  df-oprab 6089  df-mpo 6090  df-recs 6576  df-frec 6662  df-seqfrec 10885

This theorem is used by:  seqeq123d  10893  seq3f1olemqsum  10950  seqf1oglem2  10957  bcval5  11201  bcn2  11202  seq3shft  11603  iserex  12105  iser3shft  12112  isumsplit  12258  ntrivcvgap  12315  eftlub  12457  gzsumfzval  13711  gzsumval2  13714  mulgnndir  13954  gzsumshift  14149