19.33b2 - Intuitionistic Logic Explorer

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Theorem 19.33b2 1617

Description: The antecedent provides a condition implying the converse of 19.33 1472. Compare Theorem 19.33 of [Margaris] p. 90. This variation of 19.33bdc 1618 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 718	. . . . 5 $\/$ $\/$
2		alnex 1487	. . . . . 6
3		alnex 1487	. . . . . 6
4	2, 3	orbi12i 754	. . . . 5 $\/$ $\/$
5	1, 4	bitr4i 186	. . . 4 $\/$ $\/$
6		pm2.53 712	. . . . . . 7 $\/$
7	6	orcoms 720	. . . . . 6 $\/$
8	7	al2imi 1446	. . . . 5 $\/$
9		pm2.53 712	. . . . . 6 $\/$
10	9	al2imi 1446	. . . . 5 $\/$
11	8, 10	orim12d 776	. . . 4 $\/$ $\/$ $\/$
12	5, 11	syl5bi 151	. . 3 $\/$ $\/$ $\/$
13	12	com12 30	. 2 $\/$ $\/$ $\/$
14		19.33 1472	. 2 $\/$ $\/$
15	13, 14	impbid1 141	1 $\/$ $\/$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4

wb 104 $\/$ wo 698

wal 1341

wex 1480

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-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-gen 1437 ax-ie2 1482

This theorem depends on definitions: df-bi 116 df-tru 1346 df-fal 1349

This theorem is referenced by: 19.33bdc 1618

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