19.33b2 - Intuitionistic Logic Explorer

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Theorem 19.33b2 1560

Description: The antecedent provides a condition implying the converse of 19.33 1413. Compare Theorem 19.33 of [Margaris] p. 90. This variation of 19.33bdc 1561 is intuitionistically valid without a decidability condition. (Contributed by Mario Carneiro, 2-Feb-2015.)

**Assertion**
Ref	Expression
19.33b2	$\/$ $\/$ $\/$

**Proof of Theorem 19.33b2**
Step	Hyp	Ref	Expression
1		orcom 679	. . . . 5 $\/$ $\/$
2		alnex 1428	. . . . . 6
3		alnex 1428	. . . . . 6
4	2, 3	orbi12i 713	. . . . 5 $\/$ $\/$
5	1, 4	bitr4i 185	. . . 4 $\/$ $\/$
6		pm2.53 673	. . . . . . 7 $\/$
7	6	orcoms 681	. . . . . 6 $\/$
8	7	al2imi 1387	. . . . 5 $\/$
9		pm2.53 673	. . . . . 6 $\/$
10	9	al2imi 1387	. . . . 5 $\/$
11	8, 10	orim12d 732	. . . 4 $\/$ $\/$ $\/$
12	5, 11	syl5bi 150	. . 3 $\/$ $\/$ $\/$
13	12	com12 30	. 2 $\/$ $\/$ $\/$
14		19.33 1413	. 2 $\/$ $\/$
15	13, 14	impbid1 140	1 $\/$ $\/$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4

wb 103 $\/$ wo 661

wal 1282

wex 1421

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-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-gen 1378 ax-ie2 1423

This theorem depends on definitions: df-bi 115 df-tru 1287 df-fal 1290

This theorem is referenced by: 19.33bdc 1561

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