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 2802	1 ⊢ (𝜑 → seq𝑀( + , 𝐹) = seq𝑁(𝑄, 𝐺))

Syntax hints:   → wi 4   = wceq 1570  seqcseq 14039

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-xp 5669  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-iota 6494  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-seq 14040

This theorem is referenced by:  relexpsucnnr  15064  sseqval  34759  bj-finsumval0  37910  itcoval  49424