dveeq2or - Intuitionistic Logic Explorer

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Theorem dveeq2or 1830

Description: Quantifier introduction when one pair of variables is distinct. Like dveeq2 1829 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		ax12or 1522	. . . . . 6 $\/$ $\/$
2		orass 768	. . . . . 6 $\/$ $\/$ $\/$ $\/$
3	1, 2	mpbir 146	. . . . 5 $\/$ $\/$
4		pm1.4 728	. . . . . 6 $\/$ $\/$
5	4	orim1i 761	. . . . 5 $\/$ $\/$ $\/$ $\/$
6	3, 5	ax-mp 5	. . . 4 $\/$ $\/$
7		orass 768	. . . 4 $\/$ $\/$ $\/$ $\/$
8	6, 7	mpbi 145	. . 3 $\/$ $\/$
9		ax16 1827	. . . . . 6
10	9	a5i 1557	. . . . 5
11		id 19	. . . . 5
12	10, 11	jaoi 717	. . . 4 $\/$
13	12	orim2i 762	. . 3 $\/$ $\/$ $\/$
14	8, 13	ax-mp 5	. 2 $\/$
15		df-nf 1475	. . . 4
16	15	biimpri 133	. . 3
17	16	orim2i 762	. 2 $\/$ $\/$
18	14, 17	ax-mp 5	1 $\/$

Colors of variables: wff set class

Syntax hints:

wi 4 $\/$ wo 709

wal 1362

wnf 1474

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548

This theorem depends on definitions: df-bi 117 df-nf 1475 df-sb 1777

This theorem is referenced by: equs5or 1844 sbal1yz 2020 copsexg 4277