This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem cases2ALT

Description: Alternate proof of cases2 , not using dedlema or dedlemb . (Contributed by BJ, 6-Apr-2019) (Proof shortened by Wolf Lammen, 2-Jan-2020) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion cases2ALT 
|- ( ( ( ph /\ ps ) \/ ( -. ph /\ ch ) ) <-> ( ( ph -> ps ) /\ ( -. ph -> ch ) ) )

Proof

Step Hyp Ref Expression
1 pm3.4 
 |-  ( ( ph /\ ps ) -> ( ph -> ps ) )
2 pm2.24 
 |-  ( ph -> ( -. ph -> ch ) )
3 2 adantr 
 |-  ( ( ph /\ ps ) -> ( -. ph -> ch ) )
4 1 3 jca 
 |-  ( ( ph /\ ps ) -> ( ( ph -> ps ) /\ ( -. ph -> ch ) ) )
5 pm2.21 
 |-  ( -. ph -> ( ph -> ps ) )
6 5 adantr 
 |-  ( ( -. ph /\ ch ) -> ( ph -> ps ) )
7 pm3.4 
 |-  ( ( -. ph /\ ch ) -> ( -. ph -> ch ) )
8 6 7 jca 
 |-  ( ( -. ph /\ ch ) -> ( ( ph -> ps ) /\ ( -. ph -> ch ) ) )
9 4 8 jaoi 
 |-  ( ( ( ph /\ ps ) \/ ( -. ph /\ ch ) ) -> ( ( ph -> ps ) /\ ( -. ph -> ch ) ) )
10 pm2.27 
 |-  ( ph -> ( ( ph -> ps ) -> ps ) )
11 10 imdistani 
 |-  ( ( ph /\ ( ph -> ps ) ) -> ( ph /\ ps ) )
12 11 orcd 
 |-  ( ( ph /\ ( ph -> ps ) ) -> ( ( ph /\ ps ) \/ ( -. ph /\ ch ) ) )
13 12 adantrr 
 |-  ( ( ph /\ ( ( ph -> ps ) /\ ( -. ph -> ch ) ) ) -> ( ( ph /\ ps ) \/ ( -. ph /\ ch ) ) )
14 pm2.27 
 |-  ( -. ph -> ( ( -. ph -> ch ) -> ch ) )
15 14 imdistani 
 |-  ( ( -. ph /\ ( -. ph -> ch ) ) -> ( -. ph /\ ch ) )
16 15 olcd 
 |-  ( ( -. ph /\ ( -. ph -> ch ) ) -> ( ( ph /\ ps ) \/ ( -. ph /\ ch ) ) )
17 16 adantrl 
 |-  ( ( -. ph /\ ( ( ph -> ps ) /\ ( -. ph -> ch ) ) ) -> ( ( ph /\ ps ) \/ ( -. ph /\ ch ) ) )
18 13 17 pm2.61ian 
 |-  ( ( ( ph -> ps ) /\ ( -. ph -> ch ) ) -> ( ( ph /\ ps ) \/ ( -. ph /\ ch ) ) )
19 9 18 impbii 
 |-  ( ( ( ph /\ ps ) \/ ( -. ph /\ ch ) ) <-> ( ( ph -> ps ) /\ ( -. ph -> ch ) ) )