Theorem 3bitr2d 545

Description: Deduction from transitivity of biconditional.

Hypotheses

Ref	Expression
3bitr2d.1	|- (ph -> (ps <-> ch))
3bitr2d.2	|- (ph -> (th <-> ch))
3bitr2d.3	|- (ph -> (th <-> ta))

Assertion

Ref	Expression
3bitr2d	|- (ph -> (ps <-> ta))

Proof of Theorem 3bitr2d

Step	Hyp	Ref	Expression
1		3bitr2d.1	. . 3 |- (ph -> (ps <-> ch))
2		3bitr2d.2	. . 3 |- (ph -> (th <-> ch))
3	1, 2	bitr4d 530	. 2 |- (ph -> (ps <-> th))
4		3bitr2d.3	. 2 |- (ph -> (th <-> ta))
5	3, 4	bitrd 527	1 |- (ph -> (ps <-> ta))

Syntax hints:   -> wi 3   <-> wb 146

This theorem is referenced by:  lnopcon 9954  lnfncon 9981  cdj3lem1 10352  cnfilca 10545

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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