Theorem 3biant1d 1377

Description: A conjunction is equivalent to a threefold conjunction with single truth, analogous to biantrud 510. (Contributed by Alexander van der Vekens, 26-Sep-2017.)

Hypothesis

Ref	Expression
3biantd.1	|- ( ph -> th )

Assertion

Ref	Expression
3biant1d	|- ( ph -> ( ( ch /\ ps ) <-> ( th /\ ch /\ ps ) ) )

Proof of Theorem 3biant1d

Step	Hyp	Ref	Expression
1		3biantd.1	. . 3 |- ( ph -> th )
2	1	biantrurd 511	. 2 |- ( ph -> ( ( ch /\ ps ) <-> ( th /\ ( ch /\ ps ) ) ) )
3		3anass 988	. 2 |- ( ( th /\ ch /\ ps ) <-> ( th /\ ( ch /\ ps ) ) )
4	2, 3	syl6bbr 267	1 |- ( ph -> ( ( ch /\ ps ) <-> ( th /\ ch /\ ps ) ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 984

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 189  df-an 373  df-3an 986

This theorem is referenced by:  metuel2  21573  itgsubst  22994  clwlkisclwwlk  25510  itg2addnclem2  31987