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Metamath Proof Explorer

Theorem dfvd3

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

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

Proof

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