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Theorem elimif 3691

Description: Elimination of a conditional operator contained in a wff

. (Contributed by NM, 15-Feb-2005.)

**Hypotheses**
Ref	Expression
elimif.1
elimif.2

**Assertion**
Ref	Expression
elimif	$/\$ $\/$ $/\$

**Proof of Theorem elimif**
Step	Hyp	Ref	Expression
1		exmid 404	. . 3 $\/$
2	1	biantrur 492	. 2 $\/$ $/\$
3		andir 838	. 2 $\/$ $/\$ $/\$ $\/$ $/\$
4		iftrue 3668	. . . . 5
5		elimif.1	. . . . 5
6	4, 5	syl 15	. . . 4
7	6	pm5.32i 618	. . 3 $/\$ $/\$
8		iffalse 3669	. . . . 5
9		elimif.2	. . . . 5
10	8, 9	syl 15	. . . 4
11	10	pm5.32i 618	. . 3 $/\$ $/\$
12	7, 11	orbi12i 507	. 2 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
13	2, 3, 12	3bitri 262	1 $/\$ $\/$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 176 $\/$ wo 357 $/\$ wa 358

wceq 1642

cif 3662

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334

This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-if 3663

This theorem is referenced by: eqif 3695 elif 3696 ifel 3697