Theorem seqeq123d 10717

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 10714	. 2 |- ( ph -> seq M ( .+ , F ) = seq N ( .+ , F ) )
3		seqeq123d.2	. . 3 |- ( ph -> .+ = Q )
4	3	seqeq2d 10715	. 2 |- ( ph -> seq N ( .+ , F ) = seq N ( Q , F ) )
5		seqeq123d.3	. . 3 |- ( ph -> F = G )
6	5	seqeq3d 10716	. 2 |- ( ph -> seq N ( Q , F ) = seq N ( Q , G ) )
7	2, 4, 6	3eqtrd 2268	1 |- ( ph -> seq M ( .+ , F ) = seq N ( Q , G ) )

Syntax hints:   -> wi 4   = wceq 1397   seqcseq 10708

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-cnv 4733  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-recs 6470  df-frec 6556  df-seqfrec 10709

This theorem is referenced by:  igsumvalx  13471