Theorem seqeq123d 14048

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

Hypotheses

Ref	Expression
seqeq123d.1	⊢ (𝜑 → 𝑀 = 𝑁)
seqeq123d.2	⊢ (𝜑 → + = 𝑄)
seqeq123d.3	⊢ (𝜑 → 𝐹 = 𝐺)

Assertion

Ref	Expression
seqeq123d	⊢ (𝜑 → seq𝑀( + , 𝐹) = seq𝑁(𝑄, 𝐺))

Proof of Theorem seqeq123d

Step	Hyp	Ref	Expression
1		seqeq123d.1	. . 3 ⊢ (𝜑 → 𝑀 = 𝑁)
2	1	seqeq1d 14045	. 2 ⊢ (𝜑 → seq𝑀( + , 𝐹) = seq𝑁( + , 𝐹))
3		seqeq123d.2	. . 3 ⊢ (𝜑 → + = 𝑄)
4	3	seqeq2d 14046	. 2 ⊢ (𝜑 → seq𝑁( + , 𝐹) = seq𝑁(𝑄, 𝐹))
5		seqeq123d.3	. . 3 ⊢ (𝜑 → 𝐹 = 𝐺)
6	5	seqeq3d 14047	. 2 ⊢ (𝜑 → seq𝑁(𝑄, 𝐹) = seq𝑁(𝑄, 𝐺))
7	2, 4, 6	3eqtrd 2779	1 ⊢ (𝜑 → seq𝑀( + , 𝐹) = seq𝑁(𝑄, 𝐺))

Syntax hints:   → wi 4   = wceq 1537  seqcseq 14039

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-xp 5695  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-iota 6516  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-seq 14040

This theorem is referenced by:  relexpsucnnr  15061  sseqval  34370  bj-finsumval0  37268  itcoval  48511