Theorem seqeq123d 10818

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

Hypotheses

Ref	Expression
seqeq123d.1	|- ( ph -> M = N )
seqeq123d.2	|- ( ph -> .+ = Q )
seqeq123d.3	|- ( ph -> F = G )

Assertion

Ref	Expression
seqeq123d	|- ( ph -> seq M ( .+ , F ) = seq N ( Q , G ) )

Proof of Theorem seqeq123d

Step	Hyp	Ref	Expression
1		seqeq123d.1	. . 3 |- ( ph -> M = N )
2	1	seqeq1d 10815	. 2 |- ( ph -> seq M ( .+ , F ) = seq N ( .+ , F ) )
3		seqeq123d.2	. . 3 |- ( ph -> .+ = Q )
4	3	seqeq2d 10816	. 2 |- ( ph -> seq N ( .+ , F ) = seq N ( Q , F ) )
5		seqeq123d.3	. . 3 |- ( ph -> F = G )
6	5	seqeq3d 10817	. 2 |- ( ph -> seq N ( Q , F ) = seq N ( Q , G ) )
7	2, 4, 6	3eqtrd 2269	1 |- ( ph -> seq M ( .+ , F ) = seq N ( Q , G ) )

Syntax hints:   -> wi 4   = wceq 1398   seqcseq 10809

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-cnv 4757  df-dm 4759  df-rn 4760  df-res 4761  df-iota 5312  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-recs 6536  df-frec 6622  df-seqfrec 10810

This theorem is referenced by:  igsumvalx  13602