Theorem seqeq123d 10638

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 10635	. 2 |- ( ph -> seq M ( .+ , F ) = seq N ( .+ , F ) )
3		seqeq123d.2	. . 3 |- ( ph -> .+ = Q )
4	3	seqeq2d 10636	. 2 |- ( ph -> seq N ( .+ , F ) = seq N ( Q , F ) )
5		seqeq123d.3	. . 3 |- ( ph -> F = G )
6	5	seqeq3d 10637	. 2 |- ( ph -> seq N ( Q , F ) = seq N ( Q , G ) )
7	2, 4, 6	3eqtrd 2244	1 |- ( ph -> seq M ( .+ , F ) = seq N ( Q , G ) )

Syntax hints:   -> wi 4   = wceq 1373   seqcseq 10629

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-cnv 4701  df-dm 4703  df-rn 4704  df-res 4705  df-iota 5251  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-recs 6414  df-frec 6500  df-seqfrec 10630

This theorem is referenced by:  igsumvalx  13336