elim2if - Mathbox for Thierry Arnoux

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Theorem elim2if 29363

Description: Elimination of two conditional operators contained in a wff

. (Contributed by Thierry Arnoux, 25-Jan-2017.)

**Assertion**
Ref	Expression
elim2if	$/\$ $\/$ $/\$ $/\$ $\/$ $/\$

**Proof of Theorem elim2if**
Step	Hyp	Ref	Expression
1		iftrue 4092	. . 3
2		elim2if.1	. . 3
3	1, 2	syl 17	. 2
4		iffalse 4095	. . . . 5
5	4	eqeq1d 2624	. . . 4
6		elim2if.2	. . . 4
7	5, 6	syl6bir 244	. . 3
8	4	eqeq1d 2624	. . . 4
9		elim2if.3	. . . 4
10	8, 9	syl6bir 244	. . 3
11	7, 10	elimifd 29362	. 2 $/\$ $\/$ $/\$
12	3, 11	cases 992	1 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 196 $\/$ wo 383 $/\$ wa 384

wceq 1483

cif 4086

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-if 4087

This theorem is referenced by: elim2ifim 29364