Theorem seqeq2d 10676

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
seqeq2d	|- ( ph -> seq M ( A , F ) = seq M ( B , F ) )

Proof of Theorem seqeq2d

Step	Hyp	Ref	Expression
1		seqeqd.1	. 2 |- ( ph -> A = B )
2		seqeq2 10673	. 2 |- ( A = B -> seq M ( A , F ) = seq M ( B , F ) )
3	1, 2	syl 14	1 |- ( ph -> seq M ( A , F ) = seq M ( B , F ) )

Syntax hints:   -> wi 4   = wceq 1395   seqcseq 10669

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 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-cnv 4727  df-dm 4729  df-rn 4730  df-res 4731  df-iota 5278  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-recs 6451  df-frec 6537  df-seqfrec 10670

This theorem is referenced by:  seqeq123d  10678  gsumpropd  13425  gsumress  13428  mulgfvalg  13658  submmulg  13703  subgmulg  13725