Metamath Proof Explorer


Theorem e123

Description: A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Jun-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypotheses e123.1
|- (. ph ->. ps ).
e123.2
|- (. ph ,. ch ->. th ).
e123.3
|- (. ph ,. ch ,. ta ->. et ).
e123.4
|- ( ps -> ( th -> ( et -> ze ) ) )
Assertion e123
|- (. ph ,. ch ,. ta ->. ze ).

Proof

Step Hyp Ref Expression
1 e123.1
 |-  (. ph ->. ps ).
2 e123.2
 |-  (. ph ,. ch ->. th ).
3 e123.3
 |-  (. ph ,. ch ,. ta ->. et ).
4 e123.4
 |-  ( ps -> ( th -> ( et -> ze ) ) )
5 1 vd13
 |-  (. ph ,. ch ,. ta ->. ps ).
6 2 vd23
 |-  (. ph ,. ch ,. ta ->. th ).
7 5 6 3 4 e333
 |-  (. ph ,. ch ,. ta ->. ze ).