Theorem seqeq2d 13961

Description: Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)

Hypothesis

Ref	Expression
seqeqd.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
seqeq2d	⊢ (𝜑 → seq𝑀(𝐴, 𝐹) = seq𝑀(𝐵, 𝐹))

Proof of Theorem seqeq2d

Step	Hyp	Ref	Expression
1		seqeqd.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		seqeq2 13958	. 2 ⊢ (𝐴 = 𝐵 → seq𝑀(𝐴, 𝐹) = seq𝑀(𝐵, 𝐹))
3	1, 2	syl 17	1 ⊢ (𝜑 → seq𝑀(𝐴, 𝐹) = seq𝑀(𝐵, 𝐹))

Syntax hints:   → wi 4   = wceq 1547  seqcseq 13954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-xp 5624  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-iota 6441  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-seq 13955

This theorem is referenced by:  seqeq123d  13963  sadfval  16412  smufval  16437  gsumvalx  18635  gsumpropd  18637  gsumress  18641  mulgfval  19036  mulgfvalALT  19037  ressmulgnnd  19045  submmulg  19085  subgmulg  19107  dvnfval  25907