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

Step	Hyp	Ref	Expression
1		simp3 998	. . 3
2	1	a1i 11	. 2
3		simp3 998	. . 3
4		oteqex2 4744	. . 3
5	3, 4	syl5ibr 221	. 2
6		opex 4716	. . . . . . . 8
7		opthg 4727	. . . . . . . 8
8	6, 7	mpan 670	. . . . . . 7
9	8	simprbda 623	. . . . . 6
10		opeqex 4743	. . . . . 6
11	9, 10	syl 16	. . . . 5
12	4	adantl 466	. . . . 5
13	11, 12	anbi12d 710	. . . 4
14		df-3an 975	. . . 4
15		df-3an 975	. . . 4
16	13, 14, 15	3bitr4g 288	. . 3
17	16	expcom 435	. 2
18	2, 5, 17	pm5.21ndd 354	1

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