Theorem bi123imp0 38702

Description: Similar to 3imp 1256 except all implications are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)

Hypothesis

Ref	Expression
bi23imp0.1	|- ( ph <-> ( ps <-> ( ch <-> th ) ) )

Assertion

Ref	Expression
bi123imp0	|- ( ( ph /\ ps /\ ch ) -> th )

Proof of Theorem bi123imp0

Step	Hyp	Ref	Expression
1		bi23imp0.1	. . 3 |- ( ph <-> ( ps <-> ( ch <-> th ) ) )
2		biimp 205	. . . 4 |- ( ( ps <-> ( ch <-> th ) ) -> ( ps -> ( ch <-> th ) ) )
3		biimp 205	. . . 4 |- ( ( ch <-> th ) -> ( ch -> th ) )
4	2, 3	syl6 35	. . 3 |- ( ( ps <-> ( ch <-> th ) ) -> ( ps -> ( ch -> th ) ) )
5	1, 4	sylbi 207	. 2 |- ( ph -> ( ps -> ( ch -> th ) ) )
6	5	3imp 1256	1 |- ( ( ph /\ ps /\ ch ) -> th )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037

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

This theorem is referenced by: (None)