ifpimim - Mathbox for Richard Penner

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Theorem ifpimim 37854

Description: Consequnce of implication. (Contributed by RP, 17-Apr-2020.)

**Assertion**
Ref	Expression
ifpimim	if- if- if-

**Proof of Theorem ifpimim**
Step	Hyp	Ref	Expression
1		pm2.521 166	. . . . . 6
2	1	orim1i 539	. . . . 5 $\/$ $\/$
3	2	adantr 481	. . . 4 $\/$ $/\$ $\/$ $\/$
4		id 22	. . . . . 6
5	4	orci 405	. . . . 5 $\/$
6	5	a1i 11	. . . 4 $\/$ $/\$ $\/$ $\/$
7	3, 6	jca 554	. . 3 $\/$ $/\$ $\/$ $\/$ $/\$ $\/$
8	4	orci 405	. . . . 5 $\/$
9	8	a1i 11	. . . 4 $\/$ $/\$ $\/$ $\/$
10		simpr 477	. . . 4 $\/$ $/\$ $\/$ $\/$
11	9, 10	jca 554	. . 3 $\/$ $/\$ $\/$ $\/$ $/\$ $\/$
12	7, 11	jca 554	. 2 $\/$ $/\$ $\/$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$
13		pm4.81 381	. . . . 5
14	13	bicomi 214	. . . 4
15		ifpbi1 37822	. . . 4 if- if-
16	14, 15	ax-mp 5	. . 3 if- if-
17		dfifp4 1016	. . 3 if- $\/$ $/\$ $\/$
18	16, 17	bitri 264	. 2 if- $\/$ $/\$ $\/$
19		ifpim123g 37845	. 2 if- if- $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$
20	12, 18, 19	3imtr4i 281	1 if- if- if-

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 196 $\/$ wo 383 $/\$ wa 384 if-wif 1012

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-ifp 1013

This theorem is referenced by: frege58acor 38170 frege60a 38172 frege65a 38177