elimifd - Mathbox for Thierry Arnoux

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Theorem elimifd 29362

Description: Elimination of a conditional operator contained in a wff

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

**Hypotheses**
Ref	Expression
elimifd.1
elimifd.2

**Assertion**
Ref	Expression
elimifd	$/\$ $\/$ $/\$

**Proof of Theorem elimifd**
Step	Hyp	Ref	Expression
1		exmid 431	. . . 4 $\/$
2	1	biantrur 527	. . 3 $\/$ $/\$
3	2	a1i 11	. 2 $\/$ $/\$
4		andir 912	. . 3 $\/$ $/\$ $/\$ $\/$ $/\$
5	4	a1i 11	. 2 $\/$ $/\$ $/\$ $\/$ $/\$
6		iftrue 4092	. . . . 5
7		elimifd.1	. . . . 5
8	6, 7	syl5 34	. . . 4
9	8	pm5.32d 671	. . 3 $/\$ $/\$
10		iffalse 4095	. . . . 5
11		elimifd.2	. . . . 5
12	10, 11	syl5 34	. . . 4
13	12	pm5.32d 671	. . 3 $/\$ $/\$
14	9, 13	orbi12d 746	. 2 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
15	3, 5, 14	3bitrd 294	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: elim2if 29363