Theorem oteqex2 4963

Description: Equivalence of existence implied by equality of ordered triples. (Contributed by NM, 26-Apr-2015.)

Assertion

Ref	Expression
oteqex2	|- ( <. <. A , B >. , C >. = <. <. R , S >. , T >. -> ( C e. _V <-> T e. _V ) )

Proof of Theorem oteqex2

Step	Hyp	Ref	Expression
1		opeqex 4962	. 2 |- ( <. <. A , B >. , C >. = <. <. R , S >. , T >. -> ( ( <. A , B >. e. _V /\ C e. _V ) <-> ( <. R , S >. e. _V /\ T e. _V ) ) )
2		opex 4932	. . 3 |- <. A , B >. e. _V
3	2	biantrur 527	. 2 |- ( C e. _V <-> ( <. A , B >. e. _V /\ C e. _V ) )
4		opex 4932	. . 3 |- <. R , S >. e. _V
5	4	biantrur 527	. 2 |- ( T e. _V <-> ( <. R , S >. e. _V /\ T e. _V ) )
6	1, 3, 5	3bitr4g 303	1 |- ( <. <. A , B >. , C >. = <. <. R , S >. , T >. -> ( C e. _V <-> T e. _V ) )

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

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-ne 2795  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

This theorem is referenced by:  oteqex  4964