MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oteqex Unicode version

Theorem oteqex 4390
Description: Equivalence of existence implied by equality of ordered triples. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
oteqex  |-  ( <. <. A ,  B >. ,  C >.  =  <. <. R ,  S >. ,  T >.  ->  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  <->  ( R  e.  _V  /\  S  e. 
_V  /\  T  e.  _V ) ) )

Proof of Theorem oteqex
StepHypRef Expression
1 simp3 959 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  C  e.  _V )
21a1i 11 . 2  |-  ( <. <. A ,  B >. ,  C >.  =  <. <. R ,  S >. ,  T >.  ->  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  C  e.  _V ) )
3 simp3 959 . . 3  |-  ( ( R  e.  _V  /\  S  e.  _V  /\  T  e.  _V )  ->  T  e.  _V )
4 oteqex2 4389 . . 3  |-  ( <. <. A ,  B >. ,  C >.  =  <. <. R ,  S >. ,  T >.  ->  ( C  e.  _V  <->  T  e.  _V ) )
53, 4syl5ibr 213 . 2  |-  ( <. <. A ,  B >. ,  C >.  =  <. <. R ,  S >. ,  T >.  ->  ( ( R  e.  _V  /\  S  e.  _V  /\  T  e.  _V )  ->  C  e.  _V ) )
6 opex 4368 . . . . . . . 8  |-  <. A ,  B >.  e.  _V
7 opthg 4377 . . . . . . . 8  |-  ( (
<. A ,  B >.  e. 
_V  /\  C  e.  _V )  ->  ( <. <. A ,  B >. ,  C >.  =  <. <. R ,  S >. ,  T >.  <->  ( <. A ,  B >.  =  <. R ,  S >.  /\  C  =  T ) ) )
86, 7mpan 652 . . . . . . 7  |-  ( C  e.  _V  ->  ( <. <. A ,  B >. ,  C >.  =  <. <. R ,  S >. ,  T >.  <->  ( <. A ,  B >.  =  <. R ,  S >.  /\  C  =  T ) ) )
98simprbda 607 . . . . . 6  |-  ( ( C  e.  _V  /\  <. <. A ,  B >. ,  C >.  =  <. <. R ,  S >. ,  T >. )  ->  <. A ,  B >.  =  <. R ,  S >. )
10 opeqex 4388 . . . . . 6  |-  ( <. A ,  B >.  = 
<. R ,  S >.  -> 
( ( A  e. 
_V  /\  B  e.  _V )  <->  ( R  e. 
_V  /\  S  e.  _V ) ) )
119, 10syl 16 . . . . 5  |-  ( ( C  e.  _V  /\  <. <. A ,  B >. ,  C >.  =  <. <. R ,  S >. ,  T >. )  ->  (
( A  e.  _V  /\  B  e.  _V )  <->  ( R  e.  _V  /\  S  e.  _V )
) )
124adantl 453 . . . . 5  |-  ( ( C  e.  _V  /\  <. <. A ,  B >. ,  C >.  =  <. <. R ,  S >. ,  T >. )  ->  ( C  e.  _V  <->  T  e.  _V ) )
1311, 12anbi12d 692 . . . 4  |-  ( ( C  e.  _V  /\  <. <. A ,  B >. ,  C >.  =  <. <. R ,  S >. ,  T >. )  ->  (
( ( A  e. 
_V  /\  B  e.  _V )  /\  C  e. 
_V )  <->  ( ( R  e.  _V  /\  S  e.  _V )  /\  T  e.  _V ) ) )
14 df-3an 938 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  <->  ( ( A  e.  _V  /\  B  e.  _V )  /\  C  e.  _V ) )
15 df-3an 938 . . . 4  |-  ( ( R  e.  _V  /\  S  e.  _V  /\  T  e.  _V )  <->  ( ( R  e.  _V  /\  S  e.  _V )  /\  T  e.  _V ) )
1613, 14, 153bitr4g 280 . . 3  |-  ( ( C  e.  _V  /\  <. <. A ,  B >. ,  C >.  =  <. <. R ,  S >. ,  T >. )  ->  (
( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  <->  ( R  e.  _V  /\  S  e.  _V  /\  T  e.  _V ) ) )
1716expcom 425 . 2  |-  ( <. <. A ,  B >. ,  C >.  =  <. <. R ,  S >. ,  T >.  ->  ( C  e.  _V  ->  (
( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  <->  ( R  e.  _V  /\  S  e.  _V  /\  T  e.  _V ) ) ) )
182, 5, 17pm5.21ndd 344 1  |-  ( <. <. A ,  B >. ,  C >.  =  <. <. R ,  S >. ,  T >.  ->  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  <->  ( R  e.  _V  /\  S  e. 
_V  /\  T  e.  _V ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   _Vcvv 2899   <.cop 3760
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766
  Copyright terms: Public domain W3C validator