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Theorem oteqex 4745
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 998 . . 3
21a1i 11 . 2
3 simp3 998 . . 3
4 oteqex2 4744 . . 3
53, 4syl5ibr 221 . 2
6 opex 4716 . . . . . . . 8
7 opthg 4727 . . . . . . . 8
86, 7mpan 670 . . . . . . 7
98simprbda 623 . . . . . 6
10 opeqex 4743 . . . . . 6
119, 10syl 16 . . . . 5
124adantl 466 . . . . 5
1311, 12anbi12d 710 . . . 4
14 df-3an 975 . . . 4
15 df-3an 975 . . . 4
1613, 14, 153bitr4g 288 . . 3
1716expcom 435 . 2
182, 5, 17pm5.21ndd 354 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  /\wa 369  /\w3a 973  =wceq 1395  e.wcel 1818   cvv 3109  <.cop 4035
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036
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