oteqex - Metamath Proof Explorer

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Theorem oteqex 4964

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 1063	. . 3 $/\$ $/\$
2	1	a1i 11	. 2 $/\$ $/\$
3		simp3 1063	. . 3 $/\$ $/\$
4		oteqex2 4963	. . 3
5	3, 4	syl5ibr 236	. 2 $/\$ $/\$
6		opex 4932	. . . . . . . 8
7		opthg 4946	. . . . . . . 8 $/\$ $/\$
8	6, 7	mpan 706	. . . . . . 7 $/\$
9	8	simprbda 653	. . . . . 6 $/\$
10		opeqex 4962	. . . . . 6 $/\$ $/\$
11	9, 10	syl 17	. . . . 5 $/\$ $/\$ $/\$
12	4	adantl 482	. . . . 5 $/\$
13	11, 12	anbi12d 747	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
14		df-3an 1039	. . . 4 $/\$ $/\$ $/\$ $/\$
15		df-3an 1039	. . . 4 $/\$ $/\$ $/\$ $/\$
16	13, 14, 15	3bitr4g 303	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
17	16	expcom 451	. 2 $/\$ $/\$ $/\$ $/\$
18	2, 5, 17	pm5.21ndd 369	1 $/\$ $/\$ $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

cvv 3200

cop 4183

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184

This theorem is referenced by: (None)

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