Theorem seqeq2d 13919

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 13916	. 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 1541  seqcseq 13912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-xp 5627  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6255  df-iota 6444  df-fv 6496  df-ov 7357  df-oprab 7358  df-mpo 7359  df-frecs 8219  df-wrecs 8250  df-recs 8299  df-rdg 8337  df-seq 13913

This theorem is referenced by:  seqeq123d  13921  sadfval  16367  smufval  16392  gsumvalx  18588  gsumpropd  18590  gsumress  18594  mulgfval  18986  mulgfvalALT  18987  ressmulgnnd  18995  submmulg  19035  subgmulg  19057  dvnfval  25854