Theorem 3bitr3rd 218

Description: Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)

Hypotheses

Ref	Expression
3bitr3d.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))
3bitr3d.2	⊢ (𝜑 → (𝜓 ↔ 𝜃))
3bitr3d.3	⊢ (𝜑 → (𝜒 ↔ 𝜏))

Assertion

Ref	Expression
3bitr3rd	⊢ (𝜑 → (𝜏 ↔ 𝜃))

Proof of Theorem 3bitr3rd

Step	Hyp	Ref	Expression
1		3bitr3d.3	. 2 ⊢ (𝜑 → (𝜒 ↔ 𝜏))
2		3bitr3d.1	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3		3bitr3d.2	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜃))
4	2, 3	bitr3d 189	. 2 ⊢ (𝜑 → (𝜒 ↔ 𝜃))
5	1, 4	bitr3d 189	1 ⊢ (𝜑 → (𝜏 ↔ 𝜃))

Syntax hints:   → wi 4   ↔ wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  funconstss  5614  eqneg  8649  minclpr  11200  evenennn  12348  rpcxple2  13632  rpcxplt2  13633  lgslem1  13695