Theorem 3impexpbicom 1414

Description: 3impexp 1413 with biconditional consequent of antecedent that is commuted in consequent. (Contributed by Alan Sare, 31-Dec-2011.)

Assertion

Ref	Expression
3impexpbicom	|- ( ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) <-> ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) )

Proof of Theorem 3impexpbicom

Step	Hyp	Ref	Expression
1		bicom 139	. . . 4 |- ( ( th <-> ta ) <-> ( ta <-> th ) )
2		imbi2 236	. . . . 5 |- ( ( ( th <-> ta ) <-> ( ta <-> th ) ) -> ( ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) <-> ( ( ph /\ ps /\ ch ) -> ( ta <-> th ) ) ) )
3	2	biimpcd 158	. . . 4 |- ( ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) -> ( ( ( th <-> ta ) <-> ( ta <-> th ) ) -> ( ( ph /\ ps /\ ch ) -> ( ta <-> th ) ) ) )
4	1, 3	mpi 15	. . 3 |- ( ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) -> ( ( ph /\ ps /\ ch ) -> ( ta <-> th ) ) )
5	4	3expd 1202	. 2 |- ( ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) -> ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) )
6		3impexp 1413	. . . 4 |- ( ( ( ph /\ ps /\ ch ) -> ( ta <-> th ) ) <-> ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) )
7	6	biimpri 132	. . 3 |- ( ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) -> ( ( ph /\ ps /\ ch ) -> ( ta <-> th ) ) )
8	7, 1	syl6ibr 161	. 2 |- ( ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) -> ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) )
9	5, 8	impbii 125	1 |- ( ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) <-> ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 104   /\ w3a 962

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116  df-3an 964

This theorem is referenced by: (None)