Theorem syl5rbb 249

Description: A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
syl5rbb.1	|- (ph <-> ps)
syl5rbb.2	|- (ch -> (ps <-> th))

Assertion

Ref	Expression
syl5rbb	|- (ch -> (th <-> ph))

Proof of Theorem syl5rbb

Step	Hyp	Ref	Expression
1		syl5rbb.1	. . 3 |- (ph <-> ps)
2		syl5rbb.2	. . 3 |- (ch -> (ps <-> th))
3	1, 2	syl5bb 248	. 2 |- (ch -> (ph <-> th))
4	3	bicomd 192	1 |- (ch -> (th <-> ph))

Syntax hints:   -> wi 4   <-> wb 176

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 177

This theorem is referenced by:  syl5rbbr  251  csbabg  3198  uniiunlem  3354  opkelimagekg  4272  setswith  4322  fnresdisj  5194  f1oiso  5500