Theorem seqeq123d 12116

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

Hypotheses

Ref	Expression
seqeq123d.1
seqeq123d.2
seqeq123d.3

Assertion

Ref	Expression
seqeq123d

Proof of Theorem seqeq123d

Step	Hyp	Ref	Expression
1		seqeq123d.1	. . 3
2	1	seqeq1d 12113	. 2
3		seqeq123d.2	. . 3
4	3	seqeq2d 12114	. 2
5		seqeq123d.3	. . 3
6	5	seqeq3d 12115	. 2
7	2, 4, 6	3eqtrd 2502	1

Syntax hints:  ->wi 4  =wceq 1395  seqcseq 12107

This theorem is referenced by:  sseqval  28327  relexp0  29052  relexpsucr  29053  bj-finsumval0  34663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-mpt 4512  df-cnv 5012  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fv 5601  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-recs 7061  df-rdg 7095  df-seq 12108