dveeq2or - Intuitionistic Logic Explorer

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Theorem dveeq2or 1737

Description: Quantifier introduction when one pair of variables is distinct. Like dveeq2 1736 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 1438	. . . . . 6 $\/$ $\/$
2		orass 716	. . . . . 6 $\/$ $\/$ $\/$ $\/$
3	1, 2	mpbir 144	. . . . 5 $\/$ $\/$
4		pm1.4 678	. . . . . 6 $\/$ $\/$
5	4	orim1i 709	. . . . 5 $\/$ $\/$ $\/$ $\/$
6	3, 5	ax-mp 7	. . . 4 $\/$ $\/$
7		orass 716	. . . 4 $\/$ $\/$ $\/$ $\/$
8	6, 7	mpbi 143	. . 3 $\/$ $\/$
9		ax16 1734	. . . . . 6
10	9	a5i 1475	. . . . 5
11		id 19	. . . . 5
12	10, 11	jaoi 668	. . . 4 $\/$
13	12	orim2i 710	. . 3 $\/$ $\/$ $\/$
14	8, 13	ax-mp 7	. 2 $\/$
15		df-nf 1390	. . . 4
16	15	biimpri 131	. . 3
17	16	orim2i 710	. 2 $\/$ $\/$
18	14, 17	ax-mp 7	1 $\/$

Colors of variables: wff set class

Syntax hints:

wi 4 $\/$ wo 661

wal 1282

wnf 1389

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467

This theorem depends on definitions: df-bi 115 df-nf 1390 df-sb 1686

This theorem is referenced by: equs5or 1751 sbal1yz 1918 copsexg 3999