Theorem seqeq123d 10601

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 10598	. 2 |- ( ph -> seq M ( .+ , F ) = seq N ( .+ , F ) )
3		seqeq123d.2	. . 3 |- ( ph -> .+ = Q )
4	3	seqeq2d 10599	. 2 |- ( ph -> seq N ( .+ , F ) = seq N ( Q , F ) )
5		seqeq123d.3	. . 3 |- ( ph -> F = G )
6	5	seqeq3d 10600	. 2 |- ( ph -> seq N ( Q , F ) = seq N ( Q , G ) )
7	2, 4, 6	3eqtrd 2242	1 |- ( ph -> seq M ( .+ , F ) = seq N ( Q , G ) )

Syntax hints:   -> wi 4   = wceq 1373   seqcseq 10592

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-mpt 4107  df-cnv 4683  df-dm 4685  df-rn 4686  df-res 4687  df-iota 5232  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-recs 6391  df-frec 6477  df-seqfrec 10593

This theorem is referenced by:  igsumvalx  13221