dveeq2or - Intuitionistic Logic Explorer

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Theorem dveeq2or 1809

Description: Quantifier introduction when one pair of variables is distinct. Like dveeq2 1808 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 1501	. . . . . 6 $\/$ $\/$
2		orass 762	. . . . . 6 $\/$ $\/$ $\/$ $\/$
3	1, 2	mpbir 145	. . . . 5 $\/$ $\/$
4		pm1.4 722	. . . . . 6 $\/$ $\/$
5	4	orim1i 755	. . . . 5 $\/$ $\/$ $\/$ $\/$
6	3, 5	ax-mp 5	. . . 4 $\/$ $\/$
7		orass 762	. . . 4 $\/$ $\/$ $\/$ $\/$
8	6, 7	mpbi 144	. . 3 $\/$ $\/$
9		ax16 1806	. . . . . 6
10	9	a5i 1536	. . . . 5
11		id 19	. . . . 5
12	10, 11	jaoi 711	. . . 4 $\/$
13	12	orim2i 756	. . 3 $\/$ $\/$ $\/$
14	8, 13	ax-mp 5	. 2 $\/$
15		df-nf 1454	. . . 4
16	15	biimpri 132	. . 3
17	16	orim2i 756	. 2 $\/$ $\/$
18	14, 17	ax-mp 5	1 $\/$

Colors of variables: wff set class

Syntax hints:

wi 4 $\/$ wo 703

wal 1346

wnf 1453

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527

This theorem depends on definitions: df-bi 116 df-nf 1454 df-sb 1756

This theorem is referenced by: equs5or 1823 sbal1yz 1994 copsexg 4229