Theorem bj-syl66ib 32833

Description: A mixed syllogism inference derived from syl6ib 241. In addition to bj-dvelimdv1 32835, it can also shorten alexsubALTlem4 21854 (4821>4812), supsrlem 9932 (2868>2863). (Contributed by BJ, 20-Oct-2021.)

Hypotheses

Ref	Expression
bj-syl66ib.1	|- ( ph -> ( ps -> th ) )
bj-syl66ib.2	|- ( th -> ta )
bj-syl66ib.3	|- ( ta <-> ch )

Assertion

Ref	Expression
bj-syl66ib	|- ( ph -> ( ps -> ch ) )

Proof of Theorem bj-syl66ib

Step	Hyp	Ref	Expression
1		bj-syl66ib.1	. . 3 |- ( ph -> ( ps -> th ) )
2		bj-syl66ib.2	. . 3 |- ( th -> ta )
3	1, 2	syl6 35	. 2 |- ( ph -> ( ps -> ta ) )
4		bj-syl66ib.3	. 2 |- ( ta <-> ch )
5	3, 4	syl6ib 241	1 |- ( ph -> ( ps -> ch ) )

Syntax hints:   -> wi 4   <-> wb 196

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

This theorem is referenced by:  bj-dvelimdv1  32835