Theorem e123 38989

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

Hypotheses

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

Assertion

Ref	Expression
e123	|- (. ph ,. ch ,. ta ->. ze ).

Proof of Theorem e123

Step	Hyp	Ref	Expression
1		e123.1	. . 3 |- (. ph ->. ps ).
2	1	vd13 38826	. 2 |- (. ph ,. ch ,. ta ->. ps ).
3		e123.2	. . 3 |- (. ph ,. ch ->. th ).
4	3	vd23 38827	. 2 |- (. ph ,. ch ,. ta ->. th ).
5		e123.3	. 2 |- (. ph ,. ch ,. ta ->. et ).
6		e123.4	. 2 |- ( ps -> ( th -> ( et -> ze ) ) )
7	2, 4, 5, 6	e333 38960	1 |- (. ph ,. ch ,. ta ->. ze ).

Syntax hints:   -> wi 4   (.wvd1 38785   (.wvd2 38793   (.wvd3 38803

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-an 386  df-3an 1039  df-vd1 38786  df-vd2 38794  df-vd3 38806

This theorem is referenced by:  suctrALT2VD  39071