Theorem seqeq2d 14015

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 14012	. 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 1559  seqcseq 14008

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-xp 5649  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6283  df-iota 6472  df-fv 6524  df-ov 7394  df-oprab 7395  df-mpo 7396  df-frecs 8256  df-wrecs 8287  df-recs 8336  df-rdg 8375  df-seq 14009

This theorem is referenced by:  seqeq123d  14017  sadfval  16477  smufval  16502  gsumvalx  18701  gsumpropd  18703  gsumress  18707  mulgfval  19102  mulgfvalALT  19103  ressmulgnnd  19111  submmulg  19151  subgmulg  19173  dvnfval  25972