Theorem 3bitrd 422

Description: Deduction from transitivity of biconditional.

Hypotheses

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

Assertion

Ref	Expression
3bitrd	|- (ph -> (ps <-> ta ))

Proof of Theorem 3bitrd

Step	Hyp	Ref	Expression
1		3bitrd.1	. . 3 |- (ph -> (ps <-> ch))
2		3bitrd.2	. . 3 |- (ph -> (ch <-> th))
3	1, 2	bitrd 406	. 2 |- (ph -> (ps <-> th))
4		3bitrd.3	. 2 |- (ph -> (th <-> ta ))
5	3, 4	bitrd 406	1 |- (ph -> (ps <-> ta ))

Syntax hints:   -> wi 2   <-> wb 127

This theorem is referenced by:  sbcgf 1469  dedth3v 1786  elimhyp3v 1792  keephyp3v 1795  unfilem3 3440  r1pwcl 3530  atcv0eq 5767

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

This theorem depends on definitions:  df-bi 128  df-an 198

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