Theorem 3bitr2d 272

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

Hypotheses

Ref	Expression
3bitr2d.1	⊢ (φ → (ψ ↔ χ))
3bitr2d.2	⊢ (φ → (θ ↔ χ))
3bitr2d.3	⊢ (φ → (θ ↔ τ))

Assertion

Ref	Expression
3bitr2d	⊢ (φ → (ψ ↔ τ))

Proof of Theorem 3bitr2d

Step	Hyp	Ref	Expression
1		3bitr2d.1	. . 3 ⊢ (φ → (ψ ↔ χ))
2		3bitr2d.2	. . 3 ⊢ (φ → (θ ↔ χ))
3	1, 2	bitr4d 247	. 2 ⊢ (φ → (ψ ↔ θ))
4		3bitr2d.3	. 2 ⊢ (φ → (θ ↔ τ))
5	3, 4	bitrd 244	1 ⊢ (φ → (ψ ↔ τ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177

This theorem is used by:  ceqsralt  2883