Theorem seqeq123d 14133

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 14130	. 2 ⊢ (𝜑 → seq𝑀( + , 𝐹) = seq𝑁( + , 𝐹))
3		seqeq123d.2	. . 3 ⊢ (𝜑 → + = 𝑄)
4	3	seqeq2d 14131	. 2 ⊢ (𝜑 → seq𝑁( + , 𝐹) = seq𝑁(𝑄, 𝐹))
5		seqeq123d.3	. . 3 ⊢ (𝜑 → 𝐹 = 𝐺)
6	5	seqeq3d 14132	. 2 ⊢ (𝜑 → seq𝑁(𝑄, 𝐹) = seq𝑁(𝑄, 𝐺))
7	2, 4, 6	3eqtrd 2800	1 ⊢ (𝜑 → seq𝑀( + , 𝐹) = seq𝑁(𝑄, 𝐺))

This proof depends on syntax axioms:   → wi 4   = wceq 1570  seqcseq 14124

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-xp 5657  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-iota 6487  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-seq 14125

This theorem is used by:  relexpsucnnr  15158  sseqval  35003  bj-finsumval0  38174  itcoval  49717