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

Theorem oteqex 4441
 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

Proof of Theorem oteqex
StepHypRef Expression
1 simp3 959 . . 3
21a1i 11 . 2
3 simp3 959 . . 3
4 oteqex2 4440 . . 3
53, 4syl5ibr 213 . 2
6 opex 4419 . . . . . . . 8
7 opthg 4428 . . . . . . . 8
86, 7mpan 652 . . . . . . 7
98simprbda 607 . . . . . 6
10 opeqex 4439 . . . . . 6
119, 10syl 16 . . . . 5
124adantl 453 . . . . 5
1311, 12anbi12d 692 . . . 4
14 df-3an 938 . . . 4
15 df-3an 938 . . . 4
1613, 14, 153bitr4g 280 . . 3
1716expcom 425 . 2
182, 5, 17pm5.21ndd 344 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   w3a 936   wceq 1652   wcel 1725  cvv 2948  cop 3809 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815
 Copyright terms: Public domain W3C validator