Theorem s8eqd 13614

Description: Equality theorem for a length 8 word. (Contributed by Mario Carneiro, 27-Feb-2016.)

Hypotheses

Ref	Expression
s2eqd.1	|- ( ph -> A = N )
s2eqd.2	|- ( ph -> B = O )
s3eqd.3	|- ( ph -> C = P )
s4eqd.4	|- ( ph -> D = Q )
s5eqd.5	|- ( ph -> E = R )
s6eqd.6	|- ( ph -> F = S )
s7eqd.6	|- ( ph -> G = T )
s8eqd.6	|- ( ph -> H = U )

Assertion

Ref	Expression
s8eqd	|- ( ph -> <" A B C D E F G H "> = <" N O P Q R S T U "> )

Proof of Theorem s8eqd

Step	Hyp	Ref	Expression
1		s2eqd.1	. . . 4 |- ( ph -> A = N )
2		s2eqd.2	. . . 4 |- ( ph -> B = O )
3		s3eqd.3	. . . 4 |- ( ph -> C = P )
4		s4eqd.4	. . . 4 |- ( ph -> D = Q )
5		s5eqd.5	. . . 4 |- ( ph -> E = R )
6		s6eqd.6	. . . 4 |- ( ph -> F = S )
7		s7eqd.6	. . . 4 |- ( ph -> G = T )
8	1, 2, 3, 4, 5, 6, 7	s7eqd 13613	. . 3 |- ( ph -> <" A B C D E F G "> = <" N O P Q R S T "> )
9		s8eqd.6	. . . 4 |- ( ph -> H = U )
10	9	s1eqd 13381	. . 3 |- ( ph -> <" H "> = <" U "> )
11	8, 10	oveq12d 6668	. 2 |- ( ph -> ( <" A B C D E F G "> ++ <" H "> ) = ( <" N O P Q R S T "> ++ <" U "> ) )
12		df-s8 13599	. 2 |- <" A B C D E F G H "> = ( <" A B C D E F G "> ++ <" H "> )
13		df-s8 13599	. 2 |- <" N O P Q R S T U "> = ( <" N O P Q R S T "> ++ <" U "> )
14	11, 12, 13	3eqtr4g 2681	1 |- ( ph -> <" A B C D E F G H "> = <" N O P Q R S T U "> )

Syntax hints:   -> wi 4   = wceq 1483  (class class class)co 6650   ++ cconcat 13293   <"cs1 13294   <"cs7 13591   <"cs8 13592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-s1 13302  df-s2 13593  df-s3 13594  df-s4 13595  df-s5 13596  df-s6 13597  df-s7 13598  df-s8 13599

This theorem is referenced by: (None)