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Theorem seqeq2d 11818
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
StepHypRef Expression
1 seqeqd.1 . 2  |-  ( ph  ->  A  =  B )
2 seqeq2 11815 . 2  |-  ( A  =  B  ->  seq M ( A ,  F )  =  seq M ( B ,  F ) )
31, 2syl 16 1  |-  ( ph  ->  seq M ( A ,  F )  =  seq M ( B ,  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    seqcseq 11811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-cnv 4853  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-recs 6837  df-rdg 6871  df-seq 11812
This theorem is referenced by:  seqeq123d  11820  sadfval  13653  smufval  13678  gsumvalx  15507  gsumpropd  15509  gsumress  15512  mulgfval  15633  submmulg  15667  subgmulg  15700  dvnfval  21401
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