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Theorem oteqex 4259
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 957 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  C  e.  _V )
21a1i 10 . 2  |-  ( <. <. A ,  B >. ,  C >.  =  <. <. R ,  S >. ,  T >.  ->  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  C  e.  _V ) )
3 simp3 957 . . 3  |-  ( ( R  e.  _V  /\  S  e.  _V  /\  T  e.  _V )  ->  T  e.  _V )
4 oteqex2 4258 . . 3  |-  ( <. <. A ,  B >. ,  C >.  =  <. <. R ,  S >. ,  T >.  ->  ( C  e.  _V  <->  T  e.  _V ) )
53, 4syl5ibr 212 . 2  |-  ( <. <. A ,  B >. ,  C >.  =  <. <. R ,  S >. ,  T >.  ->  ( ( R  e.  _V  /\  S  e.  _V  /\  T  e.  _V )  ->  C  e.  _V ) )
6 opex 4237 . . . . . . . 8  |-  <. A ,  B >.  e.  _V
7 opthg 4246 . . . . . . . 8  |-  ( (
<. A ,  B >.  e. 
_V  /\  C  e.  _V )  ->  ( <. <. A ,  B >. ,  C >.  =  <. <. R ,  S >. ,  T >.  <->  ( <. A ,  B >.  =  <. R ,  S >.  /\  C  =  T ) ) )
86, 7mpan 651 . . . . . . 7  |-  ( C  e.  _V  ->  ( <. <. A ,  B >. ,  C >.  =  <. <. R ,  S >. ,  T >.  <->  ( <. A ,  B >.  =  <. R ,  S >.  /\  C  =  T ) ) )
98simprbda 606 . . . . . 6  |-  ( ( C  e.  _V  /\  <. <. A ,  B >. ,  C >.  =  <. <. R ,  S >. ,  T >. )  ->  <. A ,  B >.  =  <. R ,  S >. )
10 opeqex 4257 . . . . . 6  |-  ( <. A ,  B >.  = 
<. R ,  S >.  -> 
( ( A  e. 
_V  /\  B  e.  _V )  <->  ( R  e. 
_V  /\  S  e.  _V ) ) )
119, 10syl 15 . . . . 5  |-  ( ( C  e.  _V  /\  <. <. A ,  B >. ,  C >.  =  <. <. R ,  S >. ,  T >. )  ->  (
( A  e.  _V  /\  B  e.  _V )  <->  ( R  e.  _V  /\  S  e.  _V )
) )
124adantl 452 . . . . 5  |-  ( ( C  e.  _V  /\  <. <. A ,  B >. ,  C >.  =  <. <. R ,  S >. ,  T >. )  ->  ( C  e.  _V  <->  T  e.  _V ) )
1311, 12anbi12d 691 . . . 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 936 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  <->  ( ( A  e.  _V  /\  B  e.  _V )  /\  C  e.  _V ) )
15 df-3an 936 . . . 4  |-  ( ( R  e.  _V  /\  S  e.  _V  /\  T  e.  _V )  <->  ( ( R  e.  _V  /\  S  e.  _V )  /\  T  e.  _V ) )
1613, 14, 153bitr4g 279 . . 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 424 . 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 343 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649
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