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

Metamath Proof Explorer

Theorem ee223

Description: e223 without virtual deductions. (Contributed by Alan Sare, 12-Dec-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypotheses ee223.1 
|- ( ph -> ( ps -> ch ) )
ee223.2 
|- ( ph -> ( ps -> th ) )
ee223.3 
|- ( ph -> ( ps -> ( ta -> et ) ) )
ee223.4 
|- ( ch -> ( th -> ( et -> ze ) ) )
Assertion ee223 
|- ( ph -> ( ps -> ( ta -> ze ) ) )

Proof

Step Hyp Ref Expression
1 ee223.1 
 |-  ( ph -> ( ps -> ch ) )
2 ee223.2 
 |-  ( ph -> ( ps -> th ) )
3 ee223.3 
 |-  ( ph -> ( ps -> ( ta -> et ) ) )
4 ee223.4 
 |-  ( ch -> ( th -> ( et -> ze ) ) )
5 1 4 syl6 
 |-  ( ph -> ( ps -> ( th -> ( et -> ze ) ) ) )
6 5 com34 
 |-  ( ph -> ( ps -> ( et -> ( th -> ze ) ) ) )
7 6 com23 
 |-  ( ph -> ( et -> ( ps -> ( th -> ze ) ) ) )
8 7 com12 
 |-  ( et -> ( ph -> ( ps -> ( th -> ze ) ) ) )
9 3 8 syl8 
 |-  ( ph -> ( ps -> ( ta -> ( ph -> ( ps -> ( th -> ze ) ) ) ) ) )
10 9 com34 
 |-  ( ph -> ( ps -> ( ph -> ( ta -> ( ps -> ( th -> ze ) ) ) ) ) )
11 10 pm2.43a 
 |-  ( ph -> ( ps -> ( ta -> ( ps -> ( th -> ze ) ) ) ) )
12 11 com34 
 |-  ( ph -> ( ps -> ( ps -> ( ta -> ( th -> ze ) ) ) ) )
13 12 pm2.43d 
 |-  ( ph -> ( ps -> ( ta -> ( th -> ze ) ) ) )
14 13 com34 
 |-  ( ph -> ( ps -> ( th -> ( ta -> ze ) ) ) )
15 2 14 mpdd 
 |-  ( ph -> ( ps -> ( ta -> ze ) ) )