Theorem 3bitr3rd 549

Description: Deduction from transitivity of biconditional.

Hypotheses

Ref	Expression
3bitr3d.1	|- (ph -> (ps <-> ch))
3bitr3d.2	|- (ph -> (ps <-> th))
3bitr3d.3	|- (ph -> (ch <-> ta))

Assertion

Ref	Expression
3bitr3rd	|- (ph -> (ta <-> th))

Proof of Theorem 3bitr3rd

Step	Hyp	Ref	Expression
1		3bitr3d.3	. 2 |- (ph -> (ch <-> ta))
2		3bitr3d.1	. . 3 |- (ph -> (ps <-> ch))
3		3bitr3d.2	. . 3 |- (ph -> (ps <-> th))
4	2, 3	bitr3d 530	. 2 |- (ph -> (ch <-> th))
5	1, 4	bitr3d 530	1 |- (ph -> (ta <-> th))

Syntax hints:   -> wi 3   <-> wb 146

This theorem is referenced by:  sbc5g 1954  sbc6g 1955  sbccomg 1989  ltaddsubt 5631  leaddsubt 5633  climshft2 7106  efifolem5 8726  unopf1ot 9840

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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