Theorem otthg 4954

Description: Ordered triple theorem, closed form. (Contributed by Alexander van der Vekens, 10-Mar-2018.)

Assertion

Ref	Expression
otthg	|- ( ( A e. U /\ B e. V /\ C e. W ) -> ( <. A , B , C >. = <. D , E , F >. <-> ( A = D /\ B = E /\ C = F ) ) )

Proof of Theorem otthg

Step	Hyp	Ref	Expression
1		df-ot 4186	. . 3 |- <. A , B , C >. = <. <. A , B >. , C >.
2		df-ot 4186	. . 3 |- <. D , E , F >. = <. <. D , E >. , F >.
3	1, 2	eqeq12i 2636	. 2 |- ( <. A , B , C >. = <. D , E , F >. <-> <. <. A , B >. , C >. = <. <. D , E >. , F >. )
4		opex 4932	. . . . 5 |- <. A , B >. e. _V
5		opthg 4946	. . . . 5 |- ( ( <. A , B >. e. _V /\ C e. W ) -> ( <. <. A , B >. , C >. = <. <. D , E >. , F >. <-> ( <. A , B >. = <. D , E >. /\ C = F ) ) )
6	4, 5	mpan 706	. . . 4 |- ( C e. W -> ( <. <. A , B >. , C >. = <. <. D , E >. , F >. <-> ( <. A , B >. = <. D , E >. /\ C = F ) ) )
7		opthg 4946	. . . . . 6 |- ( ( A e. U /\ B e. V ) -> ( <. A , B >. = <. D , E >. <-> ( A = D /\ B = E ) ) )
8	7	anbi1d 741	. . . . 5 |- ( ( A e. U /\ B e. V ) -> ( ( <. A , B >. = <. D , E >. /\ C = F ) <-> ( ( A = D /\ B = E ) /\ C = F ) ) )
9		df-3an 1039	. . . . 5 |- ( ( A = D /\ B = E /\ C = F ) <-> ( ( A = D /\ B = E ) /\ C = F ) )
10	8, 9	syl6bbr 278	. . . 4 |- ( ( A e. U /\ B e. V ) -> ( ( <. A , B >. = <. D , E >. /\ C = F ) <-> ( A = D /\ B = E /\ C = F ) ) )
11	6, 10	sylan9bbr 737	. . 3 |- ( ( ( A e. U /\ B e. V ) /\ C e. W ) -> ( <. <. A , B >. , C >. = <. <. D , E >. , F >. <-> ( A = D /\ B = E /\ C = F ) ) )
12	11	3impa 1259	. 2 |- ( ( A e. U /\ B e. V /\ C e. W ) -> ( <. <. A , B >. , C >. = <. <. D , E >. , F >. <-> ( A = D /\ B = E /\ C = F ) ) )
13	3, 12	syl5bb 272	1 |- ( ( A e. U /\ B e. V /\ C e. W ) -> ( <. A , B , C >. = <. D , E , F >. <-> ( A = D /\ B = E /\ C = F ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   <.cotp 4185

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  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-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-ot 4186

This theorem is referenced by:  otsndisj  4979  otiunsndisj  4980  otiunsndisjX  41298