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Theorem seqeq2d 12923
Description: Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
Hypothesis
Ref Expression
seqeqd.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
seqeq2d (𝜑 → seq𝑀(𝐴, 𝐹) = seq𝑀(𝐵, 𝐹))

Proof of Theorem seqeq2d
StepHypRef Expression
1 seqeqd.1 . 2 (𝜑𝐴 = 𝐵)
2 seqeq2 12920 . 2 (𝐴 = 𝐵 → seq𝑀(𝐴, 𝐹) = seq𝑀(𝐵, 𝐹))
31, 2syl 17 1 (𝜑 → seq𝑀(𝐴, 𝐹) = seq𝑀(𝐵, 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1596  seqcseq 12916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-mpt 4838  df-xp 5224  df-cnv 5226  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-iota 5964  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-seq 12917
This theorem is referenced by:  seqeq123d  12925  sadfval  15297  smufval  15322  gsumvalx  17392  gsumpropd  17394  gsumress  17398  mulgfval  17664  submmulg  17708  subgmulg  17730  dvnfval  23805
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