Theorem dfvd3 38807

Description: Definition of a 3-hypothesis virtual deduction. (Contributed by Alan Sare, 14-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
dfvd3	|- ( (. ph ,. ps ,. ch ->. th ). <-> ( ph -> ( ps -> ( ch -> th ) ) ) )

Proof of Theorem dfvd3

Step	Hyp	Ref	Expression
1		df-vd3 38806	. 2 |- ( (. ph ,. ps ,. ch ->. th ). <-> ( ( ph /\ ps /\ ch ) -> th ) )
2		df-3an 1039	. . . . 5 |- ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ ch ) )
3	2	imbi1i 339	. . . 4 |- ( ( ( ph /\ ps /\ ch ) -> th ) <-> ( ( ( ph /\ ps ) /\ ch ) -> th ) )
4		impexp 462	. . . 4 |- ( ( ( ( ph /\ ps ) /\ ch ) -> th ) <-> ( ( ph /\ ps ) -> ( ch -> th ) ) )
5	3, 4	bitri 264	. . 3 |- ( ( ( ph /\ ps /\ ch ) -> th ) <-> ( ( ph /\ ps ) -> ( ch -> th ) ) )
6		impexp 462	. . 3 |- ( ( ( ph /\ ps ) -> ( ch -> th ) ) <-> ( ph -> ( ps -> ( ch -> th ) ) ) )
7	5, 6	bitri 264	. 2 |- ( ( ( ph /\ ps /\ ch ) -> th ) <-> ( ph -> ( ps -> ( ch -> th ) ) ) )
8	1, 7	bitri 264	1 |- ( (. ph ,. ps ,. ch ->. th ). <-> ( ph -> ( ps -> ( ch -> th ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ 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:  dfvd3i  38808  dfvd3ir  38809