Theorem seqeq123d 10876

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 10873	. 2 ⊢ (𝜑 → seq𝑀( + , 𝐹) = seq𝑁( + , 𝐹))
3		seqeq123d.2	. . 3 ⊢ (𝜑 → + = 𝑄)
4	3	seqeq2d 10874	. 2 ⊢ (𝜑 → seq𝑁( + , 𝐹) = seq𝑁(𝑄, 𝐹))
5		seqeq123d.3	. . 3 ⊢ (𝜑 → 𝐹 = 𝐺)
6	5	seqeq3d 10875	. 2 ⊢ (𝜑 → seq𝑁(𝑄, 𝐹) = seq𝑁(𝑄, 𝐺))
7	2, 4, 6	3eqtrd 2275	1 ⊢ (𝜑 → seq𝑀( + , 𝐹) = seq𝑁(𝑄, 𝐺))

Syntax hints:   → wi 4   = wceq 1402  seqcseq 10867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-mpt 4192  df-cnv 4780  df-dm 4782  df-rn 4783  df-res 4784  df-iota 5335  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-recs 6570  df-frec 6656  df-seqfrec 10868

This theorem is referenced by:  gzsumvalx  13692