Theorem seqeq123d 10389

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 10386	. 2 |- ( ph -> seq M ( .+ , F ) = seq N ( .+ , F ) )
3		seqeq123d.2	. . 3 |- ( ph -> .+ = Q )
4	3	seqeq2d 10387	. 2 |- ( ph -> seq N ( .+ , F ) = seq N ( Q , F ) )
5		seqeq123d.3	. . 3 |- ( ph -> F = G )
6	5	seqeq3d 10388	. 2 |- ( ph -> seq N ( Q , F ) = seq N ( Q , G ) )
7	2, 4, 6	3eqtrd 2202	1 |- ( ph -> seq M ( .+ , F ) = seq N ( Q , G ) )

Syntax hints:   -> wi 4   = wceq 1343   seqcseq 10380

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-cnv 4612  df-dm 4614  df-rn 4615  df-res 4616  df-iota 5153  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-recs 6273  df-frec 6359  df-seqfrec 10381

This theorem is referenced by: (None)