Theorem e333 38960

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

Hypotheses

Ref	Expression
e333.1	|- (. ph ,. ps ,. ch ->. th ).
e333.2	|- (. ph ,. ps ,. ch ->. ta ).
e333.3	|- (. ph ,. ps ,. ch ->. et ).
e333.4	|- ( th -> ( ta -> ( et -> ze ) ) )

Assertion

Ref	Expression
e333	|- (. ph ,. ps ,. ch ->. ze ).

Proof of Theorem e333

Step	Hyp	Ref	Expression
1		e333.3	. . . . . . 7 |- (. ph ,. ps ,. ch ->. et ).
2	1	dfvd3i 38808	. . . . . 6 |- ( ph -> ( ps -> ( ch -> et ) ) )
3	2	3imp 1256	. . . . 5 |- ( ( ph /\ ps /\ ch ) -> et )
4		e333.1	. . . . . . . . 9 |- (. ph ,. ps ,. ch ->. th ).
5	4	dfvd3i 38808	. . . . . . . 8 |- ( ph -> ( ps -> ( ch -> th ) ) )
6	5	3imp 1256	. . . . . . 7 |- ( ( ph /\ ps /\ ch ) -> th )
7		e333.2	. . . . . . . . 9 |- (. ph ,. ps ,. ch ->. ta ).
8	7	dfvd3i 38808	. . . . . . . 8 |- ( ph -> ( ps -> ( ch -> ta ) ) )
9	8	3imp 1256	. . . . . . 7 |- ( ( ph /\ ps /\ ch ) -> ta )
10		e333.4	. . . . . . 7 |- ( th -> ( ta -> ( et -> ze ) ) )
11	6, 9, 10	syl2im 40	. . . . . 6 |- ( ( ph /\ ps /\ ch ) -> ( ( ph /\ ps /\ ch ) -> ( et -> ze ) ) )
12	11	pm2.43i 52	. . . . 5 |- ( ( ph /\ ps /\ ch ) -> ( et -> ze ) )
13	3, 12	syl5com 31	. . . 4 |- ( ( ph /\ ps /\ ch ) -> ( ( ph /\ ps /\ ch ) -> ze ) )
14	13	pm2.43i 52	. . 3 |- ( ( ph /\ ps /\ ch ) -> ze )
15	14	3exp 1264	. 2 |- ( ph -> ( ps -> ( ch -> ze ) ) )
16	15	dfvd3ir 38809	1 |- (. ph ,. ps ,. ch ->. ze ).

Syntax hints:   -> wi 4   /\ w3a 1037   (.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-vd3 38806

This theorem is referenced by:  e33  38961  e123  38989