dveeq2or - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > dveeq2or		Unicode version

Theorem dveeq2or 1697

Description: Quantifier introduction when one pair of variables is distinct. Like dveeq2 1696 but connecting

by a disjunction rather than negation and implication makes the theorem stronger in intuitionistic logic. (Contributed by Jim Kingdon, 1-Feb-2018.)

**Assertion**
Ref	Expression
dveeq2or	$\/$

**Proof of Theorem dveeq2or**
Step	Hyp	Ref	Expression
1		ax-i12 1398	. . . . . 6 $\/$ $\/$
2		orass 684	. . . . . 6 $\/$ $\/$ $\/$ $\/$
3	1, 2	mpbir 134	. . . . 5 $\/$ $\/$
4		pm1.4 646	. . . . . 6 $\/$ $\/$
5	4	orim1i 677	. . . . 5 $\/$ $\/$ $\/$ $\/$
6	3, 5	ax-mp 7	. . . 4 $\/$ $\/$
7		orass 684	. . . 4 $\/$ $\/$ $\/$ $\/$
8	6, 7	mpbi 133	. . 3 $\/$ $\/$
9		ax16 1694	. . . . . 6
10	9	a5i 1435	. . . . 5
11		id 19	. . . . 5
12	10, 11	jaoi 636	. . . 4 $\/$
13	12	orim2i 678	. . 3 $\/$ $\/$ $\/$
14	8, 13	ax-mp 7	. 2 $\/$
15		df-nf 1350	. . . 4
16	15	biimpri 124	. . 3
17	16	orim2i 678	. 2 $\/$ $\/$
18	14, 17	ax-mp 7	1 $\/$

Colors of variables: wff set class

Syntax hints:

wi 4 $\/$ wo 629

wal 1241

wnf 1349

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427

This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646

This theorem is referenced by: equs5or 1711 sbal1yz 1877 copsexg 3981