Theorem 3bitr3d

Description: Deduction from transitivity of biconditional. Useful for converting conditional definitions in a formula. (Contributed by NM, 24-Apr-1996)

	Ref	Expression
Hypotheses	3bitr3d.1	|- ( ph -> ( ps <-> ch ) )
	3bitr3d.2	|- ( ph -> ( ps <-> th ) )
	3bitr3d.3	|- ( ph -> ( ch <-> ta ) )
Assertion	3bitr3d	|- ( ph -> ( th <-> ta ) )

Proof

Step	Hyp	Ref	Expression
1		3bitr3d.1	|- ( ph -> ( ps <-> ch ) )
2		3bitr3d.2	|- ( ph -> ( ps <-> th ) )
3		3bitr3d.3	|- ( ph -> ( ch <-> ta ) )
4	2 1	bitr3d	|- ( ph -> ( th <-> ch ) )
5	4 3	bitrd	|- ( ph -> ( th <-> ta ) )