Theorem syl5rbb 533

Description: A syllogism inference from two biconditionals.

Hypotheses

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

Assertion

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

Proof of Theorem syl5rbb

Step	Hyp	Ref	Expression
1		syl5rbb.1	. . 3 |- (ph -> (ps <-> ch))
2		syl5rbb.2	. . 3 |- (th <-> ps)
3	1, 2	syl5bb 532	. 2 |- (ph -> (th <-> ch))
4	3	bicomd 521	1 |- (ph -> (ch <-> th))

Syntax hints:   -> wi 3   <-> wb 146

This theorem is referenced by:  syl5rbbr 535  sbcralt 1990  sbcralgf 1992  fnresdisj 3597  f1oiso 3904  rdglim2 3949  2ndconst 4097  1idpr 5133  infmsup 6068  fz1sbct 6517  isph 8481

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 147  df-an 225

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