Theorem syl5rbbr 251

Description: A syllogism inference from two biconditionals. (Contributed by NM, 25-Nov-1994.)

Hypotheses

Ref	Expression
syl5rbbr.1	|- (ps <-> ph)
syl5rbbr.2	|- (ch -> (ps <-> th))

Assertion

Ref	Expression
syl5rbbr	|- (ch -> (th <-> ph))

Proof of Theorem syl5rbbr

Step	Hyp	Ref	Expression
1		syl5rbbr.1	. . 3 |- (ps <-> ph)
2	1	bicomi 193	. 2 |- (ph <-> ps)
3		syl5rbbr.2	. 2 |- (ch -> (ps <-> th))
4	2, 3	syl5rbb 249	1 |- (ch -> (th <-> ph))

This proof depends on syntax axioms:   -> wi 4   <-> wb 176

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177

This theorem is used by:  sbco3  2088  sbal2  2134  dmfco  5382  fressnfv  5440  eluniima  5470  txpcofun  5804  brfullfung  5866